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

%I A339418 #8 Dec 03 2020 18:21:29
%S A339418 1,0,1,0,1,2,1,4,2,6,9,8,20,16,35,44,55,102,105,196,242,344,540,652,
%T A339418 1084,1380,2037,2964,3912,6042,7976,11776,16634,22968,33963,46156,
%U A339418 67457,94510,133180,192316,266514,385338,540138,767008,1094576,1534704,2200821,3094248
%N A339418 Number of compositions (ordered partitions) of n into an even number of squares.
%H A339418 <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%H A339418 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%F A339418 G.f.: 4 / (3 + 2 * theta_3(x) - theta_3(x)^2), where theta_3() is the Jacobi theta function.
%F A339418 a(n) = (A006456(n) + A317665(n)) / 2.
%F A339418 a(n) = Sum_{k=0..n} A006456(k) * A317665(n-k).
%e A339418 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].
%p A339418 b:= proc(n, t) option remember; local r, f, g;
%p A339418       if n=0 then t else r, f, g:=$0..2; while f<=n
%p A339418       do r, f, g:= r+b(n-f, 1-t), f+2*g-1, g+1 od; r fi
%p A339418     end:
%p A339418 a:= n-> b(n, 1):
%p A339418 seq(a(n), n=0..50);  # _Alois P. Heinz_, Dec 03 2020
%t A339418 nmax = 47; CoefficientList[Series[4/(3 + 2 EllipticTheta[3, 0, x] - EllipticTheta[3, 0, x]^2), {x, 0, nmax}], x]
%Y A339418 Cf. A000290, A006456, A034008, A317665, A339364, A339416, A339419.
%K A339418 nonn
%O A339418 0,6
%A A339418 _Ilya Gutkovskiy_, Dec 03 2020