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 1-3 of 3 results.

A303667 Expansion of 2/((1 - x)*(3 - theta_3(x))), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 13, 18, 25, 36, 52, 74, 104, 147, 209, 297, 421, 596, 845, 1199, 1701, 2411, 3417, 4844, 6868, 9738, 13806, 19573, 27749, 39342, 55778, 79079, 112112, 158944, 225342, 319479, 452941, 642152, 910404, 1290719, 1829911, 2594344, 3678108, 5214606, 7392970, 10481335
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 28 2018

Keywords

Comments

Partial sums of A006456.

Links

Crossrefs

Cf. A000290, A006456, A010052, A302833, A303668.

Programs

  • Maple
    b:= proc(n) option remember;
      `if`(n=0, 1, add(b(n-i^2), i=1..isqrt(n)))
    end:
a:= proc(n) option remember;
      `if`(n<0, 0, b(n)+a(n-1))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Apr 28 2018
  • Mathematica
    nmax = 45; CoefficientList[Series[2/((1 - x) (3 - EllipticTheta[3, 0, x])), {x, 0, nmax}], x]
nmax = 45; CoefficientList[Series[1/((1 - x) (1 - Sum[x^k^2, {k, 1, nmax}])), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Boole[IntegerQ[k^(1/2)]] a[n - k], {k, 1, n}]; Accumulate[Table[a[n], {n, 0, 45}]]

Formula

G.f.: 1/((1 - x)*(1 - Sum_{k>=1} x^(k^2))).

A303908 Expansion of 1/(2 + x - theta_2(sqrt(x))/(2*x^(1/8))), where theta_2() is the Jacobi theta function.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 2, 0, 0, 3, 1, 0, 5, 2, 0, 9, 5, 0, 15, 10, 1, 27, 20, 3, 46, 40, 9, 80, 78, 22, 139, 152, 51, 242, 290, 114, 427, 550, 247, 753, 1034, 525, 1340, 1933, 1092, 2396, 3602, 2237, 4312, 6685, 4519, 7813, 12380, 9027, 14239, 22877, 17866, 26110, 42214, 35072, 48123, 77829, 68379
Offset: 0

Views

Author

Ilya Gutkovskiy, May 02 2018

Keywords

Comments

Number of compositions (ordered partitions) of n into triangular numbers > 1.

Links

Crossrefs

Cf. A000217, A023361, A212804, A280542, A303668, A303906, A303907.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<0, 0, `if`(n=0, 1,
      add(a(n-j*(j+1)/2), j=2..isqrt(2*n))))
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, May 02 2018
  • Mathematica
    nmax = 62; CoefficientList[Series[1/(2 + x - EllipticTheta[2, 0, Sqrt[x]]/(2 x^(1/8))), {x, 0, nmax}], x]
nmax = 62; CoefficientList[Series[1/(1 - Sum[x^(k (k + 1)/2), {k, 2, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[SquaresR[1, 8 k + 1] a[n - k], {k, 2, n}]/2; Table[a[n], {n, 0, 62}]

Formula

G.f.: 1/(1 - Sum_{k>=2} x^(k*(k+1)/2)).

A303905 Expansion of (1/(1 - x))*Product_{k>=1} (1 + x^(k*(k+1)/2)).

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 9, 10, 10, 11, 12, 13, 15, 16, 17, 19, 20, 22, 24, 24, 26, 29, 30, 31, 34, 36, 37, 41, 44, 44, 47, 50, 52, 56, 59, 62, 65, 67, 70, 73, 75, 79, 85, 89, 91, 96, 100, 102, 108, 113, 116, 123, 129, 132, 137, 142, 147, 153, 158, 162, 169, 176, 182, 190, 196, 201
Offset: 0

Views

Author

Ilya Gutkovskiy, May 02 2018

Keywords

Comments

Partial sums of A024940.

Links

Crossrefs

Cf. A000217, A024940, A036469, A038348, A061208, A248801, A302835, A303668.

Programs

  • Mathematica
    nmax = 69; CoefficientList[Series[1/(1 - x) Product[1 + x^(k (k + 1)/2), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ exp(3 * Pi^(1/3) * ((sqrt(2) - 1) * Zeta(3/2))^(2/3) * n^(1/3) / 2^(4/3)) / (2^(1/3) * (sqrt(2) - 1)^(1/3) * sqrt(3) * Pi^(2/3) * Zeta(3/2)^(1/3) * n^(1/6)). - Vaclav Kotesovec, May 04 2018
Showing 1-3 of 3 results.

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