cp's OEIS Frontend

A224366 Number of compositions of n^2 into sums of squares.

%I A224366 #16 Jun 09 2018 04:37:11
%S A224366 1,1,2,11,124,2870,133462,12477207,2344649612,885591183971,
%T A224366 672331353833716,1025954712063362545,3146790000180780110540,
%U A224366 19400015532276248131470280,240398159948843792847457589388,5987629866666297470033540284817068,299759874416459708067727376075503706332
%N A224366 Number of compositions of n^2 into sums of squares.
%C A224366 Equals the row sums of triangle A232266.
%H A224366 Alois P. Heinz, <a href="/A224366/b224366.txt">Table of n, a(n) for n = 0..81</a>
%F A224366 a(n) = [x^(n^2)] 1/(1 - Sum_{k>=1} x^(k^2)).
%F A224366 a(n) = A006456(n^2).
%F A224366 a(n) = Sum_{k=1..n} A006456(n^2-k^2) for n>=1 with a(0)=1.
%e A224366 Illustrate a(n) = Sum_{k=1..n} A006456(n^2-k^2):
%e A224366 a(1) = 1 = 1;
%e A224366 a(2) = 2 = 1 + 1;
%e A224366 a(3) = 11 = 7 + 3 + 1;
%e A224366 a(4) = 124 = 88 + 30 + 5 + 1;
%e A224366 a(5) = 2870 = 2024 + 710 + 124 + 11 + 1;
%e A224366 a(6) = 133462 = 94137 + 33033 + 5767 + 502 + 22 + 1;
%e A224366 a(7) = 12477207 = 8800750 + 3088365 + 539192 + 46832 + 2024 + 43 + 1; ...
%p A224366 b:= proc(n) option remember; local i; if n=0 then 1
%p A224366       else 0; for i while i^2<=n do %+b(n-i^2) od fi
%p A224366     end:
%p A224366 a:= n-> b(n^2):
%p A224366 seq(a(n), n=0..17);  # _Alois P. Heinz_, Aug 12 2017
%t A224366 b[0] = 1; b[n_] := b[n] = Sum[b[n-k], {k, Select[Range[n], IntegerQ[ Sqrt[#]]&]}];
%t A224366 a[n_] := b[n^2];
%t A224366 Table[a[n], {n, 0, 16}] (* _Jean-François Alcover_, Jun 09 2018 *)
%o A224366 (PARI) {a(n)=polcoeff(1/(1-sum(k=1,n,x^(k^2))+x*O(x^(n^2))),n^2)}
%o A224366 for(n=0,21,print1(a(n),", "))
%o A224366 (PARI) {A006456(n)=polcoeff(1/(1-sum(k=1,sqrtint(n+1),x^(k^2))+x*O(x^n)),n)}
%o A224366 {a(n)=if(n==0,1,sum(k=1,n,A006456(n^2-k^2)))}
%o A224366 for(n=0,21,print1(a(n),", "))
%Y A224366 Cf. A006456, A232266.
%K A224366 nonn
%O A224366 0,3
%A A224366 _Paul D. Hanna_, Apr 05 2013