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.

A303668 Expansion of 1/((1 - x)*(2 - theta_2(sqrt(x))/(2*x^(1/8)))), where theta_2() is the Jacobi theta function.

Original entry on oeis.org

1, 2, 3, 5, 8, 12, 19, 30, 46, 71, 111, 172, 266, 413, 640, 991, 1537, 2383, 3692, 5722, 8869, 13745, 21303, 33018, 51172, 79308, 122917, 190503, 295251, 457597, 709207, 1099165, 1703546, 2640245, 4091988, 6341979, 9829132, 15233702, 23609994, 36592010, 56712212, 87895562
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 28 2018

Keywords

Comments

Partial sums of A023361.

Links

Crossrefs

Cf. A000217, A010054, A023361, A302835, A303667.

Programs

  • Maple
    b:= proc(n) option remember; `if`(n=0, 1,
      add(`if`(issqr(8*j+1), b(n-j), 0), j=1..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 = 41; CoefficientList[Series[1/((1 - x) (2 - EllipticTheta[2, 0, Sqrt[x]]/(2 x^(1/8)))), {x, 0, nmax}], x]
nmax = 41; CoefficientList[Series[1/((1 - x) (1 - Sum[x^(k (k + 1)/2), {k, 1, nmax}])), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[SquaresR[1, 8 k + 1] a[n - k], {k, 1, n}]/2; Accumulate[Table[a[n], {n, 0, 41}]]

Formula

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

A303909 Expansion of 2*(1 - x)/(3 - theta_3(x)), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 1, 1, 2, 4, 5, 6, 8, 13, 19, 26, 36, 51, 74, 105, 148, 208, 296, 421, 597, 846, 1198, 1699, 2409, 3417, 4843, 6865, 9732, 13799, 19566, 27739, 39325, 55749, 79041, 112063, 158877, 225241, 319331, 452734, 641866, 910001, 1290137, 1829079, 2593169, 3676457, 5212266
Offset: 0

Views

Author

Ilya Gutkovskiy, May 02 2018

Keywords

Comments

First differences of A006456.

Links

Crossrefs

Cf. A000290, A006456, A078134, A303667.

Programs

  • Maple
    b:= proc(n) option remember; `if`(n<0, 0,
      `if`(n=0, 1, add(b(n-j^2), j=1..isqrt(n))))
    end:
a:= n-> b(n)-`if`(n=0, 0, b(n-1)):
seq(a(n), n=0..60);  # Alois P. Heinz, May 02 2018
  • Mathematica
    nmax = 50; CoefficientList[Series[2 (1 - x)/(3 - EllipticTheta[3, 0, x]), {x, 0, nmax}], x]
nmax = 50; CoefficientList[Series[(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}]; Differences[Table[a[n], {n, -1, 50}]]

Formula

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

A304633 Expansion of 2/((1 - x)*(3 + 2*x - theta_3(x))), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 4, 4, 5, 7, 7, 7, 9, 12, 13, 13, 16, 20, 23, 23, 27, 35, 41, 42, 47, 61, 71, 75, 82, 104, 124, 134, 146, 178, 217, 237, 258, 307, 377, 419, 456, 535, 651, 739, 804, 933, 1126, 1300, 1422, 1629, 1955, 2275, 2513, 2846, 3397, 3972, 4435, 4990, 5904
Offset: 0

Views

Author

Ilya Gutkovskiy, May 15 2018

Keywords

Comments

Partial sums of A280542.

Links

Crossrefs

Cf. A000290, A001156, A006456, A010052, A248801, A280542, A302833, A303667.

Programs

  • Mathematica
    nmax = 62; CoefficientList[Series[2/((1 - x) (3 + 2 x - EllipticTheta[3, 0, x])), {x, 0, nmax}], x]
nmax = 62; CoefficientList[Series[1/((1 - x) (1 - Sum[x^k^2, {k, 2, nmax}])), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Boole[IntegerQ[k^(1/2)] && k != 1] a[n - k], {k, 1, n}]; Accumulate[Table[a[n], {n, 0, 62}]]

Formula

G.f.: 1/((1 - x)*(1 - Sum_{k>=2} x^(k^2))).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.