A339418 Number of compositions (ordered partitions) of n into an even number of squares.

1, 0, 1, 0, 1, 2, 1, 4, 2, 6, 9, 8, 20, 16, 35, 44, 55, 102, 105, 196, 242, 344, 540, 652, 1084, 1380, 2037, 2964, 3912, 6042, 7976, 11776, 16634, 22968, 33963, 46156, 67457, 94510, 133180, 192316, 266514, 385338, 540138, 767008, 1094576, 1534704, 2200821, 3094248

Offset: 0

Author

Ilya Gutkovskiy, Dec 03 2020

Keywords

• nonn

Examples

			a(9) = 6 because we have [4, 1, 1, 1, 1, 1], [1, 4, 1, 1, 1, 1], [1, 1, 4, 1, 1, 1], [1, 1, 1, 4, 1, 1], [1, 1, 1, 1, 4, 1] and [1, 1, 1, 1, 1, 4].
		

Links

• Index entries for sequences related to compositions
• Index entries for sequences related to sums of squares

Crossrefs

Cf. A000290, A006456, A034008, A317665, A339364, A339416, A339419.

Programs

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, 1):
seq(a(n), n=0..50);  # Alois P. Heinz, Dec 03 2020

Mathematica

nmax = 47; CoefficientList[Series[4/(3 + 2 EllipticTheta[3, 0, x] - EllipticTheta[3, 0, x]^2), {x, 0, nmax}], x]

Formula

G.f.: 4 / (3 + 2 * theta_3(x) - theta_3(x)^2), where theta_3() is the Jacobi theta function.

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

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