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

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

0,2

Comments

Partial sums of A006456.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Index entries for sequences related to compositions

Index entries for sequences related to sums of squares

Formula

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

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}]]

Crossrefs

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

Sequence in context: A239551 A219282 A098578 * A050811 A076968 A238430

Adjacent sequences: A303664 A303665 A303666 * A303668 A303669 A303670

Keyword

nonn

Author

Ilya Gutkovskiy, Apr 28 2018

Status

approved