A339419 Number of compositions (ordered partitions) of n into an odd number of squares.

0, 1, 0, 1, 1, 1, 3, 1, 5, 5, 7, 14, 10, 27, 27, 44, 69, 73, 144, 158, 260, 366, 466, 775, 940, 1490, 2031, 2803, 4264, 5551, 8460, 11525, 16399, 23864, 32435, 47981, 66005, 94701, 135072, 187999, 272678, 379095, 543626, 769490, 1083788, 1553661, 2177681, 3113333

Offset

0,7

Links

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

Index entries for sequences related to compositions

Index entries for sequences related to sums of squares

Formula

G.f.: 1 / (3 - theta_3(x)) - 1 / (1 + theta_3(x)), where theta_3() is the Jacobi theta function.

a(n) = (A006456(n) - A317665(n)) / 2.

a(n) = -Sum_{k=0..n-1} A006456(k) * A317665(n-k).

Example

a(9) = 5 because we have [9], [4, 4, 1], [4, 1, 4], [1, 4, 4] and [1, 1, 1, 1, 1, 1, 1, 1, 1].

Maple

b:= proc(n, t) option remember; local r, f, g;
      if n=0 then t else r, f, g:=$0..2; while f<=n
      do r, f, g:= r+b(n-f, 1-t), f+2*g-1, g+1 od; r fi
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..50);  # Alois P. Heinz, Dec 03 2020

Mathematica

nmax = 47; CoefficientList[Series[1/(3 - EllipticTheta[3, 0, x]) - 1/(1 + EllipticTheta[3, 0, x]), {x, 0, nmax}], x]

Crossrefs

Cf. A000290, A006456, A166444, A317665, A339365, A339417, A339418.

Sequence in context: A213612 A186754 A124741 * A213952 A205873 A220479

Adjacent sequences: A339416 A339417 A339418 * A339420 A339421 A339422

Keyword

nonn

Author

Ilya Gutkovskiy, Dec 03 2020

Status

approved