cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A292464 a(n) = A292136(n)^2 + A292137(n)^2.

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 5, 5, 4, 4, 4, 10, 13, 13, 13, 20, 37, 37, 29, 40, 65, 72, 74, 89, 125, 178, 196, 200, 234, 325, 410, 394, 421, 617, 772, 829, 905, 1125, 1445, 1625, 1700, 2045, 2650, 3077, 3293, 3698, 4658, 5540, 5941, 6605, 7880, 9553, 10817, 11785, 13625
Offset: 0

Views

Author

Seiichi Manyama, Sep 17 2017

Keywords

Links

Crossrefs

Cf. A278401, A278402, A278420, A292136, A292137, A292138.

Programs

  • Mathematica
    Abs[CoefficientList[Series[1/QPochhammer[I*x, x], {x, 0, 100}], x]]^2 (* Vaclav Kotesovec, Sep 17 2017 *)

Formula

a(n) = A292136(n)^2 + A292138(n)^2.
a(n) = (A278401(n)^2 + A278402(n)^2)/2.

A292137 G.f.: Im(1/(i*x; x)_inf), where (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).

Original entry on oeis.org

0, 1, 1, 0, 0, 0, -1, -2, -2, -2, -2, -3, -3, -2, -2, -2, -1, 1, 2, 2, 4, 6, 7, 8, 10, 13, 14, 14, 15, 17, 17, 15, 15, 16, 14, 10, 8, 6, 1, -5, -10, -14, -21, -31, -38, -43, -53, -64, -71, -77, -86, -97, -104, -108, -115, -124, -127, -125, -127, -130, -125, -116
Offset: 0

Views

Author

Seiichi Manyama, Sep 09 2017

Keywords

Examples

			Product_{k>=1} 1/(1 - i*x^k) = 1 + (0+1i)*x + (-1+1i)*x^2 + (-1+0i)*x^3 + (-1+0i)*x^4 + (-1+0i)*x^5 + (-2-1i)*x^6 + (-1-2i)*x^7 + ...

Links

Crossrefs

Cf. A000108, A010815, A027193, A063834, A067659, A081362, A099323, A196545, A220418, A290261, A292042, A292043, A292136, A292138, A298118, A300355.

Programs

  • Maple
    N:= 100:
S := convert(series( add( (-1)^n*x^(2*n+1)/(mul(1 - x^k,k = 1..2*n+1)), n = 0..N ), x, N+1 ), polynom):
seq(coeff(S, x, n), n = 0..N); # Peter Bala, Jan 15 2021
  • Mathematica
    Im[CoefficientList[Series[1/QPochhammer[I*x, x], {x, 0, 100}], x]] (* Vaclav Kotesovec, Sep 17 2017 *)

Formula

1/(i*x; x)_inf is the g.f. for A292136(n) + i*a(n).
a(n) = Sum (-1)^((k - 1)/2) where the sum is over all integer partitions of n into an odd number of parts and k is the number of parts. - Gus Wiseman, Mar 08 2018
G.f.: Sum_{n >= 0} (-1)^n * x^(2*n+1)/Product_{k = 1..2*n+1} (1 - x^k). - Peter Bala, Jan 15 2021

A292138 G.f.: Im(1/(-i*x; x)_inf), where (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).

Original entry on oeis.org

0, -1, -1, 0, 0, 0, 1, 2, 2, 2, 2, 3, 3, 2, 2, 2, 1, -1, -2, -2, -4, -6, -7, -8, -10, -13, -14, -14, -15, -17, -17, -15, -15, -16, -14, -10, -8, -6, -1, 5, 10, 14, 21, 31, 38, 43, 53, 64, 71, 77, 86, 97, 104, 108, 115, 124, 127, 125, 127, 130, 125, 116, 110, 103, 89
Offset: 0

Views

Author

Seiichi Manyama, Sep 09 2017

Keywords

Examples

			Product_{k>=1} 1/(1 + i*x^k) = 1 + (0-1i)*x + (-1-1i)*x^2 + (-1+0i)*x^3 + (-1+0i)*x^4 + (-1+0i)*x^5 + (-2+1i)*x^6 + (-1+2i)*x^7 + ...

Crossrefs

Cf. A292042, A292052, A292136, A292137.

Formula

1/(-i*x; x)_inf is the g.f. for A292136(n) + i*a(n).
a(n) = -A292137(n).

A293268 G.f.: Re(1/(1 + i*x/(1 + i*x^2/(1 + i*x^3/(1 + i*x^4/(1 + i*x^5/(1 + ...))))))), a continued fraction, where i is the imaginary unit.

Original entry on oeis.org

1, 0, -1, -1, 1, 3, 2, -2, -7, -6, 4, 16, 14, -9, -37, -33, 20, 87, 82, -41, -201, -198, 85, 465, 475, -178, -1084, -1150, 353, 2511, 2767, -684, -5810, -6633, 1287, 13463, 15923, -2222, -31119, -38130, 3356, 71838, 91138, -3595, -165763, -217705, -1761, 381895, 519284, 27984, -878685
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 04 2017

Keywords

Examples

			G.f. A(x) = Sum_{n>=0} (a(n) + i*A293269(n))*x^n = 1 - i*x - x^2 - (1 - i)*x^3 + (1 + 2*i)*x^4 + 3*x^5 + (2 - 3*i)*x^6 - (2 + 5*i)*x^7 - (7 + i)*x^8 - ...

Links

Crossrefs

Cf. A007325, A278399, A292136, A293269.

Programs

  • Mathematica
    nmax = 50; Re[CoefficientList[Series[1/(1 + ContinuedFractionK[I x^k, 1, {k, 1, nmax}]), {x, 0, nmax}], x]]
nmax = 50; Re[CoefficientList[Series[Sum[I^k x^(k (k + 1)) / Product[1 - x^m, {m, 1, k}], {k, 0, nmax}] / Sum[I^k x^(k^2) / Product[1 - x^m, {m, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]]

Formula

G.f.: Re( (Sum_{k>=0} i^k*x^(k*(k+1))/Product_{m=1..k} (1 - x^m)) / (Sum_{k>=0} i^k*x^(k^2)/Product_{m=1..k} (1 - x^m)) ), where i is the imaginary unit.
Showing 1-4 of 4 results.

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