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

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

0,9

Comments

First differences of A006456.

Links

Table of n, a(n) for n=0..50.

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

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}]]

Crossrefs

Cf. A000290, A006456, A078134, A303667.

Sequence in context: A245319 A037081 A270877 * A110277 A325680 A176654

Adjacent sequences: A303906 A303907 A303908 * A303910 A303911 A303912

Keyword

nonn

Author

Ilya Gutkovskiy, May 02 2018

Status

approved