My OEIS entries
Stan Sykora's contributions to the Online Encyclopedia of Integer Sequences
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The integer sequences and the decimal expansions of math constants that I have registered in OEIS fall into several generic groups: (a) novel entries stemming from my own recreational and/or professional math 'endeavors', (b) entries intended simply to complete specific families (such as platonic solids data) on OEIS, (c) entries related to physics | engineering | systems of units, (d) potential entries met casually in my readings, (e) kind of pun (such as A246917). At the beginning, many were related to my Math Constants page (filling in what looked to me as open gaps) but that is of less importance today.

Approximate chronological listing, with the most recent entries listed first:

  • A290409, A290410: Real and imaginary parts of (j+(j+(j+...)^j)^j)^j, respectively.
    Also, the unique invariant point in C of the mapping M(z)=(j+z)^j.
  • A290408: Real part of (j+(j+(j+...)^(-j))^(-j))^(-j). The corresponding imaginary part is -1/2.
    Also, the unique invariant point in C of the mapping M(z)=(j+z)^(-j).
  • (E(|x|^p))^(1/p), with x being a normally distributed random variable:
    A289090 for p = 3,  and A289091 for p =5.
  • Recurrence a(n+2) = Sum_{k=0..n}(binomial(n,k)*a(k)a(n+1-k)) with various starting values:
    Reference: Sequences related to the differential equation f'' = af'f.
    A289064, a(0) = 1, a(1) = -1
    A289065, a(0) = 2, a(1) = -1
    A289066, a(0) = 3, a(1) = +1
    A289067, a(0) = 3, a(1) = -1
    A289068, a(0) = 1, a(1) = -2
    A289069, a(0) = 3, a(1) = -2
    A289070, a(0) = 0, a(1) = +3
    A289082, a(0) = 1, a(1) = i, real parts
    A289083, a(0) = 1, a(1) = i, imaginary parts
    A289084, a(0) = 2, a(1) = i, real parts
    A289085, a(0) = 2, a(1) = i, imaginary parts
    A289086, a(0) = 1, a(1) = 2i, real parts
    A289087, a(0) = 1, a(1) = 2i, imaginary parts
    A289088, a(0) = 2, a(1) = 2i, real parts
    A289089, a(0) = 2, a(1) = 2i, imaginary parts
  • A279927: Expansion of exponential generating function arctan(x)*exp(x).
  • Number of balanced seatings of n persons at two labeled round tables.
    A277876: n!/(m*(n-m)) with m=floor(n/2).
  • A fixed point of the mapping +exp(z) in C congruent with branch K=1 of log(z)+2πKi:
    A277681 (real part), A277682 (imaginary part), A277683 (modulus). This FP is denoted as z3.
  • A fixed point of the mapping -exp(z) in C congruent with branch K=1 of log(z)+2πKi:
    A276759 (real part), A276760 (imaginary part), A276761 (modulus). This FP is denoted as z2.
  • One of a unique pair of conjugate fixed points of the logarithmic integral li(z) in C:
    A276762 (real part), A276763, (imaginary part).
  • A276710: Composite numbers m such that Product(k=0,m)C(m,k) is divisible by m^(m-1).
    Conjecture: m is prime iff Product(k=0,m)C(m,k) is divisible by m^(m-1) but not by m^m.
  • A276709: Derivative of li(x) at its positive real root. Equals 1/log(Soldner's constant).
  • A275975: Value of the Sumk=0,1,2,... (-)k/2^(2^k), related to the Jeffrey's sequence.
  • A binary sequence due to Harold Jeffreys:
    A275973 (the sequence), A275974 (its partial sums).
  • A273621: Solid angle subtended by a cone having the magic angle as its polar angle.
  • A273580: Infinite nested radical √(F0+√(F1+√(F2+...))), with Fn being the Fermat numbers.
  • Infinite nested power (1+(1+(1+...)^i)^i)^i. Invariant point of M(z)=(1+z)^i in C:
    A272875 (real part), A272876 (imaginary part), A272877 (modulus).
  • A272874: Infinite nested redical √(-1+√(+1+√(-1+√(+1+...)))).
  • A272873: Quadratic mean of 1 and π.
  • Edge lengths of some constructible regular n-gons with unit circumradius:
    A272534 (15-gon), A272535 (16-gon), A272536 (20-gon).
  • Edge lengths of some nonconstructible regular n-gons with unit circumradius:
    A272487 (heptagon), A272488 (9-gon), A272489 (11-gon), A272490 (13-gon), A272491 (19-gon).
  • A272408: Hausdorff dimension of the Rauzy fractal boundary.
  • Riemann ζ(x) function: location and value of the first local maximum for real x < 0:
    A271855, location of x1. A271856, ζ(x1).
  • Riemann zeta function at -3/2:
    A271853 (value), A271854 (derivative)
  • A271834: Newton-like formula for (1+1)^n but with reduced binomials:
    a(n) = 2^n - Summ=0..n binomial(n/GCD(n,m), m/GCD(n,m)). Note: a(n) is 0 iff n is prime.
  • A272031: Hausdorff dimension of the Heighway-Harter dragon curve boundary.
  • Probability that a random real number is evil:
    A271880, full value. A271881, difference between full value and 1/5.
  • A268508: (π/8)√3, the atomic packing factor of bcc lattices.
  • A271533: η(-1), value of the Dirichlet eta function at -1.
  • Derivative of the Dirichlet eta function at i (the imaginary unit):
    A271525, real(η'(i)). A271526, -imag(η'(i)).
  • Dirichlet eta function at i (the imaginary unit):
    A271523, real(η(i)). A271524, imag(η(i)).
  • Derivative of the Riemann zeta function at i, the imaginary unit:
    A271521, real(ζ'(i)). A271522, -imag(ζ'(i)).
  • A271452: Hoffman's approximation to π, relating it to Googol (absolute error 5e-11).
  • A270230: 3/(4π). Maximum volume fraction for a triangular prism enclosed by a sphere.
  • A268604: 6/31/4. Smallest possible perimeter index for triangles and sets of triangles.
    The index is P/sqrt(A), with P being the perimeter of a planar figure, and A the enclosed area.
  • A268487: An aspect of the Thomson problem in Physics: Number of equal electric charges for which the minimum-potential dislocation on a sphere has nonzero sum of position vectors.
  • A267413: Numbers such that dropping any binary digit gives a prime.
    Curiosity: 57735 (binary 1110000110000111) is the only such number so far with 5 binary runs -
    and a palindrome as well. The entry contains PARI code to compute such sequences in any base.
  • Fibonacci-like sequences with pseudo-randomness introduced via Gray coding:
    A265385, a(n) = gray(a(n-1)+a(n-2)).
    A265386, a(n) = gray( gray(a(n-1))+gray(a(n-2)) ).
    A265387, a(n) = gray(a(n-1))+gray(a(n-2)).
  • A266046: Real part of Q^n, where Q is the quaternion 2+j+k, with a general exposition.
  • A261813: Lower bound on the discriminant of a number field of order N, for N=3.
    The lower bound formula is (π/4)N(NN/N!)2 (referenced to Companion to Mathematics).
  • A263210, A263211: Real & imag parts of the continued fraction i/(π+i/(π+i/(...))), respectively.
  • A263208, A263209: Real & imag parts of the continued fraction i/(e+i/(e+i/(...))), respectively.
  • A263357: Solution of x*(log(x)-1)/(log(x)+1) = 1. Related to entry A263356.
  • A263356: Solution of (x-1)/(x+1) = exp(-x). Related to shared tangents of y=exp(x) and y=log(x).
  • A260816: Largest integer m such that e^m < C(n), the n-th Catalan number.
  • A260800: Value of the continued fraction π/(e+π/(e+π/(...))) = (sqrt(e^2+4*π)-e)/2.
    Also: the positive solution of x*(x+e) = π. Also: the unique attractor of the mapping M(x) = π/(e+x).
  • A260799: Value of the continued fraction e/(π+e/(π+e/(...))) = (sqrt(π^2+4*e)-π)/2.
    Also: the positive solution of x*(x+π) = e. Also: the unique attractor of the mapping M(x) = e/(π+x).
  • A257945: Common absolute value of i/(i+i/(i+i/(...))) and i/(1+i/(1+i/(...))).
  • A257896: Fraction of the horizon hidden by the curve y=ex when viewed from the origin.
  • Some special points of the logarithmic integral function li(z):
    A257817, the real part of li(i).
    A257818, the imaginary part of li(i).
    A257819, the real part of li(-1).
    A257820, the imaginary part of li(-1+ε*i) for positive ε→0.
    A257821, the unique negative real root r of Real(li(x)).
    A257822, the imaginary part of li(r+ε*i) for positive ε→0.
  • A258428: atan(1/e), slope of the line passing through origin and kissing the function y(x) = log(x).
  • A257777: atan(e), slope of the line passing through origin and kissing the function y(x) = exp(x).
  • Instances of (e/n) n :
    A257775 (n=2) and A257776 (n=3).
  • A257535: Imaginary part of -E1(i), where E1(z) is the exponential integral.
  • A257530: Value of sqrt(π/sqrt(e)) = Integral[-∞..+∞](exp(-x2)cos(k.x))=sqrt(π/exp(k2/2)), for k=1.
  • The absolute maximum of hsinc(x) = (1-cos(x))/x, the Hilbert transform of sinc(x):
    A257451 (location xmax) and A257452 (value at xmax).
  • Number of p-th power nonresidues modulo n:
    A257301 (p=3, cubic nonresidues),  A257302 (p=4),  A257303 (p=5).
  • Values of the Borwein-Borwein function I3(u,v) = Integral[0,∞](x/((u3+x3)*(v3+x3)2)1/3) :
    A257096 (u=1, v=2),  A257097 (u=2, v=1).
  • Areas of regular polygons which were not yet covered by prior entries:
    A256853 (9-gon) and A256854 (11-gon).
  • Zeeman catastrophe machine constants:
    A256719 location of near bifurcation cusp,
    A256720 location of far bifurcation cusp.
  • A256502: Largest integer not exceeding the harmonic mean of the first n squares.
  • These constants appear in formulas related to Planck's Radiation Law:
    A256500  Decimal expansion of the positive solution to x = 2*(1-exp(-x)).
    A256501  Decimal expansion of the positive solution to x = 4*(1-exp(-x)).
  • A256460: Decimal expansion of 1/273.16,
    the fraction of the triple point temperature of water corresponding to 1 K (kelvin).
  • A254756: Numbers whose hexadecimal prefixes and suffixes are all primes.
  • A254755: Left-truncatable composites: every decimal suffix is a composite number.
  • Natural numbers with property X, such that, in base 10,
    all their proper prefixes and suffixes represent numbers with property Y:
    A254750   X = any, Y = composite,
    A254751   X = any, Y = prime,
    A254752   X = composite, Y = composite,
    A254753   X = composite, Y = prime,
    A254754   X = prime, Y = composite,
  • Numbers a(n) such that m = a(n)\b is not coprime to n and,
    if m is nonzero, it also belongs to the same sequence (a truncatable property):
    A250039 (b=16), A250041 (b=10), A250043 (b=9), A250045 (b=8),
    A250047 (b=7),   A250049 (b=6),   A250051 (b=5), A250037 (b=4).
  • Numbers a(n) such that m = a(n)\b is coprime to n and,
    if m is nonzero, it also belongs to the same sequence (a truncatable property):

    A250038 (b=16), A250040 (b=10), A250042 (b=9), A250044 (b=8),
    A250046 (b=7),   A250048 (b=6),   A250050 (b=5), A250036 (b=4).
  • A250034: Numerators a(n) of a rational-valued function s(n)
    related to certain coprime sequence densities. The corresponding denominators are in A072155.
  • A250031: a(n) is the numerator of the density of naturals m such that gcd(m,m\n)=1.
    A250032: a(n) is the numerator of the density of naturals m such that gcd(m,m\n)>1.
    A250033: a(n) gives the denominators for A250031(n) as well as A250032(n).
  • Numbers m that are:
    A248499 coprime to floor(m/10), A248500 not coprime to floor(m/10),
    A248501 coprime to floor(m/16), A248502 not coprime to floor(m/16),
  • A249103: Ratio of the blackbody radiation constant to the fine structure constant.
  • A248748: Number of rooted binary trees with n leaves and bichromatic internal vertices.
  • A246917: Number of letters in the n-th word of the Czech national anthem, Kde domov muj.
  • Some special values of the Dedekind eta function η(x.i), for:
    A248190 (x = 1/2), A248191 (x = 2), A248192 (x = 4)
  • A248177: Real part of the digamma function psi(i), i being the imaginary unit.
  • A248176: Value of the digamma function psi(x) for x = -1/2.
  • A247446: The atomic packing factor of the diamond crystal lattice, π*sqrt(3)/16.
  • A247445: Negative derivative of Dawson integral at its inflection points.
  • A247412: The tetrahedral angle (in degrees).
  • A246499: ζ(2)/eγ, γ being the Euler-Mascheroni constant.
  • A246131: Composite numbers m such that c(m) - 2 = 0 mod m,
    where c(m) stands for the central binomial coefficient or, alternatively, for the Catalan number.
  • Values of binomial(2n,n)-2 mod np:
    A246130 (p=1), A246132 (p=2), A246133 (p=3), A246134 (p=4)
  • The smallest number m, such that
    A245509: the first odd number after n^m is composite.
    A245510: records in A245509.
    A245511: the largest odd number < n^m is not prime.
    A245512: records in A245511.
    A245513: neither of the two odd numbers which bracket n^m is a prime.
    A245514: at least one of the two odd numbers which bracket n^m is not a prime.
  • A245087: a(n) is the largest number m, such that 2^m is a divisor of (n!)!
  • A245080: Numbers m such that omega(m) is a proper divisor of bigomega(m).
  • A244499: e/γ, the ratio of Euler number and Euler-Mascheroni constant.
  • A244274: e*γ, the product of Euler number and Euler-Mascheroni constant.
  • Coefficients & terms of some binomial identities derived from an Abel's master identity:
    A244116 & A244117, a decomposition of 1.
    A244118 & A244119, a decomposition of 1.
    A244120 & A244121, a decomposition of n^n.
    A244122 & A244123, a decomposition of n^n.
    A244124 & A244125, a decomposition of 2^n-1.
    A244126 & A244127, a decomposition of 2^n-1.
    A244128 & A244129, a decomposition of 0.
    A244130 & A244131, a decomposition of n.
    A244132 & A244133, a decomposition of n.
    A244134 & A244135, a decomposition of n^n.
    A244136 & A244137, a decomposition of n^n.
    A244138 & A244139, a decomposition of n*(n-1).
    A244140 & A244141, a decomposition of n*(-1)^n.
    A244142 & A244143, a decomposition of n.
  • Solid angle of an equilateral spherical triangle with a side length of 1 radian:
    A243710 (in steradians), A243711 (as fraction of full solid angle).
  • Solid angle subtended by a cone with polar angle of 0.0001 arcseconds:
    A243598 (current resolution limit of astronomy, expressed as fraction of full solid angle).
  • Solid angle subtended by a cone with polar angle of 1 radian:
    A243596 (in steradians), A243597 (as fraction of full solid angle).
  • Unsigned Stirling numbers of the first kind:
    A243569 s(n,8), A243570 s(n,9).
  • A243445:
    Polar angle of the cone circumscribed to a regular dodecahedron from one of its vertices.
  • Values of k^(1/sqrt(k)):
    A243444 k = 6.
    A243443 k = 7; the maximum among all k.
    A243406 k = 8.
  • A243405: Minimum among the numbers p^(n/p), where p is a prime factor of n.
  • A243203: Terms of a decomposition of NN.
  • A243202: Coefficients of a binomiaal decomposition of NN.
  • Number of cyclic arrangements of a set S such that all neighbor pairs have a property P:
    A242519 S={1,2,3,...,n}, difference Δ=2^k for some k≥0
    A242520 S={1,2,3,...,n}, difference Δ=3^k for some k≥0
    A242521 S={1,2,3,...,n}, difference Δ=b^k for some b>1 and k>1,
    A242522 S={1,2,3,...,n}, difference Δ≥2
    A242523 S={1,2,3,...,n}, difference Δ≥3
    A242524 S={1,2,3,...,n}, difference Δ≥4
    A242525 S={1,2,3,...,n}, difference Δ≤3
    A242526 S={1,2,3,...,n}, difference Δ≤4
    A242527 S={0,1,2,...,n-1}, sum Σ is prime
    A242528 S={0,1,2,...,n-1}, difference Δ and sum Σ are prime
    A242529 S={1,2,3,...,n}, neighbors are coprime
    A242530 S={1,2,3,...,n}, neighbors differ in one bit in their binary expansions
    A242531 S={1,2,3,...,n}, difference Δ is divisor of sum Σ
    A242532 S={2,3,4,...,n+1}, difference Δ is greater than 1, and a divisor of sum Σ
    A242533 S={1,2,3,...,n}, difference Δ is coprime to the sum Σ
    A242534 S={1,2,3,...,n}, difference Δ is not coprime to the sum Σ
  • Sums of the series -Sum[k=1,2,...]((-1)^k/p(k)^x), where p(k) is the k-th prime:
    A242301 (x = 2), A242302 (x = 3), A242303 (x = 4), A242304 (x = 5).
  • A242220: (10^(1/3)-1)/2, an approximation to Euler-Mascheroni constant
    (from Fauro et al, Manuale di Matematica).
  • A239798: Radius of the midsphere in a regular dodecahedron with unit edges.
  • Probability that a normal-error variable
    exceeds the mean by more than n standard deviations:

    A239382 (n = 1), A239383 (n = 2), A239384 (n = 3),
    A239385 (n = 4), A239386 (n = 5), A239387 (n = 6)
  • A238857: m-digit right-truncatable reversible primes in base n. TArray: row n lists the counts.
  • A238856: Number of digits of the largest right-truncatable reversible prime in base n.
  • A238855: Number of all right-truncatable reversible primes in base n.
  • A238854: Largest right-truncatable reversible prime in base n.
  • Right-truncatable reversible primes in some bases:
    A238850 (base 10), A238851 (base 16), A238852 (base 100), A238853 (base 256)
  • A238813: Numerators of the coefficients of Euler-Ramanujan's harmonic number expansion into negative powers of a triangular number..
  • A238238 and A238239: Polar angle of the cone cutting the full solid angle in golden ratio
    (radians and degrees, respectively).
  • Right truncatable numbers divisible by a square (in base 10):
    A237607 (the main sequence; probably infinite),
    A237608 (numbers of n-digit members).
  • A237603: Radius of the inscribed sphere in a regular dodecahedron with unit edge.
  • Right truncatable primes in base 16:
    A237600 (all 414 that exist),
    A237601 (number of n-digit members),
    A237602 (largest among the n-digit members).
  • A228613 (with Maximilian F. Hasler): Largest prime factor of (2n+1)^2 + 2..
  • A237185 and A237186: Real and imaginary parts of e^(i/π), respectively.
  • Canonical Gray cycles of length 2n:
    A236602 (counting),
    A236603 (list of lowest cycles, one for each n).
  • Vertex solid angles (steradians) in regular Platonic solids:
    A236555 (tetrahedron),
    A236556 (octahedron),
    A236557 (icosahedron),
    A236558 (dodecahedron).
  • A236367: Dihedral angle in a regular icosahedron (radians).
  • A236100 and A236101: Real and imaginary parts of π^(i/π), respectively.
  • A236098 and A236099: Real and imaginary parts of -π^(i*π), respectively.
  • A233593: √n with aperiodic Blazys expansions √n = c(1)+c(1)/(c(2)+c(2)/(c(3)+...)).
  • A233592: √n with periodic Blazys expansion √n = c(1)+c(1)/(c(2)+c(2)/(c(3)+...)).
  • Continued fractions a1+a1/(a2+a2/(a3+a3/(a4...))) for some sequencies {a1, a2, a3,...}:
    A233588 (primes; Blazys constant),
    A233589 (factorial),
    A233590 (powers of two),
    A233591 (squares),
  • Blazy's expansions of some irrational numbers:
    A233582 (π), A233583 (e), A233584 (√e), A233585 (1/γ), A233586 (2γ), A233587 (√7),
  • A232570: Replace the smallest prime factor p in n (if any) with the prime following p.
  • A232511: Replace the largest prime factor p>2 in n (if any) with the prime preceeding p.
  • Surface indices of various bodies:
    A232808 (3D sphere),
    A232809 (regular icosahedron),
    A232810 (regular dodecahedron),
    A232811 (regular octahedron),
    A232812 (regular tetrahedron),
    A232813 (closed cylinder, minimum),
    A232814 (half-open cylinder, minimum),
    A232815 (closed cone, minimum),
    A232816 (open cone, minimum),
    A232817 (open-tube).
  • A232738: Imaginary part of i^(1/8), or sin(Pi/16).
  • A232737: Real part of i^(1/9), or cos(Pi/18).
  • A232736: Imaginary part of i^(1/7), or sin(Pi/14).
  • A232735: Real part of i^(1/7), or cos(Pi/14).
  • Value of 2^n*Sum[k=0..n] kp qk for various p,q:
    A232604 (p=3, q=-1/2), A232603 (p=2, q=-1/2).
  • Value of Sum[k=0..n] kp qk for various p,q:
    A232602 (p=3, q=-2), A232601 (p=2, q=-2), A232600 (p=1, q=-2), A232599 (p=3, q=-1).
  • A232451: Prime divisors count for the Belphegor number B(n)=(10^(n+3)+666)*10^(n+1)+1.
  • A232450: Largest prime factor of the Belphegor number B(n)=(10^(n+3)+666)*10^(n+1)+1.
  • A232448: Numbers k such that A232449(k) is prime (a Belphegor prime, see the links).
    Beware of 1000000000000066600000000000001, haha.
  • A232449: The palindromic Belphegor numbers: (10^(n+3)+666)*10^(n+1)+1.
  • A231987: Side length (in radians) of a spherical square whose solid angle is one steradian.
  • A231986: Solid angle (in steradians) subtended by a spherical square of one radian side.
  • A231985: Side length (in degrees) of the spherical square whose solid angle is one deg^2.
  • A231984: Solid angle (in deg^2) of a spherical square having sides of one degree.
  • A231983: Solid angle (in sr) of a spherical square having sides of one degree.
  • A231982: One deg^2, expressed in steradians (sr).
  • A231981: One steradian (sr) expressed in deg^2.
  • A231863: 1/√(2π)
  • A231738: (1/π)^(1/e)
  • A231737: log(π)/π
  • A231736: π.log(π)
  • A231535: (π^4)/15
  • A231532, A231533: and A231534:
    Real & imaginary parts, and the absolute value, of expim(1,1), respectively.
  • A231530 and A231531:
    Real and imaginary parts, respectively, of factim(1) = Productk=1..n(k+i)
  • Recurrences a(n) = a(n-2) + n^M for various M:
    A231303 (M = 4), A231304 (M = 5), A231305 (M = 6), A231306 (M = 7),
    A231307 (M = 8), A231308 (M = 9), A231309 (M = 10).
  • Power towers of:
    A231098 (the ratio π/e),
    A231097 (the ratio e/π),
    A231096 (the inverse golden ratio φ),
    A231095 (Euler-Mascheroni constant gamma).
  • Physics constants which were assigned immutable reference values in the SI system:
    A213610 (characteristic impedance of vacuum in SI units),
    A213611 (standard atmosphere in SI units),
    A213612 (Julian year in SI seconds),
    A213613 (Gregorian year in SI seconds),
    A213614 (light year in meters).

    General context: Within current metrological systems (SI + IAU definitions), several physics constants have been "assigned" immutable values. They thus became metrological reference points, no longer subject to experimental assessment. These may not be confused with "conventional" values of some empirical quantities (such as Josephson's constant) used in applied metrology, but not assigned, and therefore subject to possible future revisions. More ....

    For OEIS it is proper to list the assigned metrological constants and some of their combinations, including: speed of light (A003678) and its square (A182999), magnetic permeability of vacuum (A019694), electric permittivity of vacuum (A081799), standard gravity in (A072915), and the five constants listed above, all in basic SI units.

  • q-Quantum transitions in systems of N >= q spin 1/2 particles, in columns by combination indices: A213343 (q = 1), A213344 (q = 2), A213345 (q = 3), A213346 (q = 4), A213347 (q = 5),
    A213348 (q = 6), A213349 (q = 7), A213350 (q = 8), A213351 (q = 9), A213352 (q = 10).

    General discussion: Consider the 2^N numbers with N-digit binary expansion. Let a pair (v,w), here called a "transition", be such that there are exactly k+q digits which are '0' in v and '1' in w, and exactly k digits which are '1' in v and '0' in w. Then T(q;N,k) is the number of all such pairs. More ...

    For given N and q, the rows of the triangle T(q;N,k) sum up to Sumk T(q;N,k) = C(2N,N-q) which is the total number of q-quantum transitions or, equivalently, the number of pairs in which the sum of binary digits of w exceeds that of v by exactly q.

    The terminology stems from the mapping of the i-th digit onto quantum states of the i-th particle (-1/2 for digit '0', +1/2 for digit '1'), the numbers onto quantum states of the system, and the pairs onto quantum transitions between states. In magnetic resonance (NMR) the most intense transitions are the single-quantum ones (q=1) with k=0, called "main transitions", while those with k>0, called "combination transitions", tend to be weaker. Zero-, double- and, in general, q-quantum transitions are detectable by special techniques.

    Specific cases: Each individual sequence covers q-quantum transitions for one value of q.
    It lists the flattened triangle T(q;N,k), with rows N = 1,2,... and columns k = 0..floor((N-q)/2).

    The case q=0 (zero-quantum transitions) is covered by the prior sequence A051288.

  • A213421: Real parts of the coefficient of Q^n, Q being the quaternion 2+i+j+k
  • A213055: Second Chandrasekhar's nearest-neighbor constant c
  • A213054: First Chandrasekhar's nearest-neighbor constant c
  • A213053: Absolute minimum of sinc(x) = sin(x)/x (negated)
  • A213007: Brun's quadruple primes constant
  • Polylogarithms li(-n,-p/q) mutiplied by ((p+q)^(n+1))/q, for small negative rationals:
    A212846 (-1/2), signs sequence in A210245 (see also A210244);
    A210246 (-1/3), signs sequence in A210247;
    A213127 (-1/4), A213128 (-1/5), A213129 (-1/6), A213130 (-1/7),
    A213131 (-1/8), A213132 (-1/9), A213133 (-1/10);
    A212847 (-2/3), A213134 (-2/5), A213135 (-2/7), A213136 (-2/9);
    A213137 (-3/4), A213138 (-3/5), A213139 (-3/7), A213140 (-3/8), A213141 (-3/10);
    A213142 (-4/5), A213143 (-4/7), A213144 (-4/9);
    A213145 (-5/6), A213146 (-5/7), A213147 (-5/8), A213148 (-5/9);
    A213149 (-6/7); A213150 (-7/8), A213151 (-7/9), A213152 (-7/10);
    A213153 (-8/9); A213154 (-9/10);
    A213155 (-1/100), A213156 (-1/1000), A213157 (-99/100):
  • A212877 (real(i!)), A212878 (imag(i!)), A212879 (|i!|), A212880 (arg(i!)):
    Real and imaginary part of i-factorial (i!) and its absolute value and argument, respectively.
  • A212792: Product of all primes in the interval ((n+1)/2,n]
  • A212791: Central binomial coefficient cb(n) purged of all primes exceeding (n+1)/2
  • A212697, A212698, A212699, A212700, A212701, A212702, A212703, A212704:
    Main transitions in systems of n particles with spin 1, 3/2, 2, 5/2, 3, 7/2, 4, 9/2, respectively.

    Consider the set S of all b^n numbers which have n digits in base b. Define as "main transition" a pair (x,y) of elements of S such that x and y differ in base b in only one digit which in y exceeds that in x by 1. These sequences give the number of such transitions for the cases b = 3, 4, 5, 6, 7, 8, 9, 10. The case b=2 (S=1/2) is covered by the prior sequence A001787

    The terminology originates from quantum theory of coupled spin systems (such as in magnetic resonance) with n particles, each with spin S = (b-1)/2. Then the i-th digit's value in base b can be intended as a label for the b = 2S+1 quantum states of the i-th particle. The most intense main quantum transitions then correspond to the above definition. Due to continuity, the correspondence holds regardless of how strongly coupled are the particles among themselves.

  • A211113: -zeta(-1/2)
  • A212480: Argument of infinite power tower of  i
  • A212479: Absolute value of infinite power tower of  i
  • A212436 and A212437: Real and imaginary parts of e^(i/e).
  • A182168: Imaginary part of i^(1/4)
  • A212223: First a(n) > 1 which have the same sum of digits in all prime bases from 2 to p(n)
  • A212284: First a(n) > 1 such that its sum of digits is the same in base 10 as in base n
  • A212283: First a(n) > 1 such that its sum of digits is the same in base 2 as in base n
  • A182587: The only solution of x^(4/x^2) = x-1
  • A212222: The sum of digits of a(n) in base b is the same for every prime b up to 11
  • A212225: (2/π)log(Φ), the exponential rate factor of golden spiral
  • A212224: Φ^(2/π), the base of the golden spiral
  • A211883, A211884: Real and imaginary parts of -(i^e), respectively.
  • A211269: Integral of a sinc-shaped peak with unit height and unit half-height width
  • A211268: Integral of a Gaussian peak with unit height and unit half-height width
  • A182007: 2*sin(π/5); the 'associate' of the golden ratio
  • A210244: Numerators of the polylogarithm li(-n,-1/2)/2
  • A208745: The gravitoid constant


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