cp's OEIS Frontend

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

%I A339419 #9 Dec 11 2020 08:48:04
%S A339419 0,1,0,1,1,1,3,1,5,5,7,14,10,27,27,44,69,73,144,158,260,366,466,775,
%T A339419 940,1490,2031,2803,4264,5551,8460,11525,16399,23864,32435,47981,
%U A339419 66005,94701,135072,187999,272678,379095,543626,769490,1083788,1553661,2177681,3113333
%N A339419 Number of compositions (ordered partitions) of n into an odd number of squares.
%H A339419 Alois P. Heinz, <a href="/A339419/b339419.txt">Table of n, a(n) for n = 0..5000</a>
%H A339419 <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%H A339419 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%F A339419 G.f.: 1 / (3 - theta_3(x)) - 1 / (1 + theta_3(x)), where theta_3() is the Jacobi theta function.
%F A339419 a(n) = (A006456(n) - A317665(n)) / 2.
%F A339419 a(n) = -Sum_{k=0..n-1} A006456(k) * A317665(n-k).
%e A339419 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].
%p A339419 b:= proc(n, t) option remember; local r, f, g;
%p A339419       if n=0 then t else r, f, g:=$0..2; while f<=n
%p A339419       do r, f, g:= r+b(n-f, 1-t), f+2*g-1, g+1 od; r fi
%p A339419     end:
%p A339419 a:= n-> b(n, 0):
%p A339419 seq(a(n), n=0..50);  # _Alois P. Heinz_, Dec 03 2020
%t A339419 nmax = 47; CoefficientList[Series[1/(3 - EllipticTheta[3, 0, x]) - 1/(1 + EllipticTheta[3, 0, x]), {x, 0, nmax}], x]
%Y A339419 Cf. A000290, A006456, A166444, A317665, A339365, A339417, A339418.
%K A339419 nonn
%O A339419 0,7
%A A339419 _Ilya Gutkovskiy_, Dec 03 2020