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

0,6

Links

Table of n, a(n) for n=0..47.

Index entries for sequences related to compositions

Index entries for sequences related to sums of squares

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).

Example

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].

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]

Crossrefs

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

Sequence in context: A205685 A334404 A143375 * A074364 A256610 A276055

Adjacent sequences: A339415 A339416 A339417 * A339419 A339420 A339421

Keyword

nonn

Author

Ilya Gutkovskiy, Dec 03 2020

Status

approved