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A339419
Number of compositions (ordered partitions) of n into an odd number of squares.
4
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
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]
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Dec 03 2020
STATUS
approved