A224366 - OEIS

%I #16 Jun 09 2018 04:37:11

%S 1,1,2,11,124,2870,133462,12477207,2344649612,885591183971,

%T 672331353833716,1025954712063362545,3146790000180780110540,

%U 19400015532276248131470280,240398159948843792847457589388,5987629866666297470033540284817068,299759874416459708067727376075503706332

%N Number of compositions of n^2 into sums of squares.

%C Equals the row sums of triangle A232266.

%H Alois P. Heinz, <a href="/A224366/b224366.txt">Table of n, a(n) for n = 0..81</a>

%F a(n) = [x^(n^2)] 1/(1 - Sum_{k>=1} x^(k^2)).

%F a(n) = A006456(n^2).

%F a(n) = Sum_{k=1..n} A006456(n^2-k^2) for n>=1 with a(0)=1.

%e Illustrate a(n) = Sum_{k=1..n} A006456(n^2-k^2):

%e a(1) = 1 = 1;

%e a(2) = 2 = 1 + 1;

%e a(3) = 11 = 7 + 3 + 1;

%e a(4) = 124 = 88 + 30 + 5 + 1;

%e a(5) = 2870 = 2024 + 710 + 124 + 11 + 1;

%e a(6) = 133462 = 94137 + 33033 + 5767 + 502 + 22 + 1;

%e a(7) = 12477207 = 8800750 + 3088365 + 539192 + 46832 + 2024 + 43 + 1; ...

%p b:= proc(n) option remember; local i; if n=0 then 1

%p       else 0; for i while i^2<=n do %+b(n-i^2) od fi

%p     end:

%p a:= n-> b(n^2):

%p seq(a(n), n=0..17);  # _Alois P. Heinz_, Aug 12 2017

%t b[0] = 1; b[n_] := b[n] = Sum[b[n-k], {k, Select[Range[n], IntegerQ[ Sqrt[#]]&]}];

%t a[n_] := b[n^2];

%t Table[a[n], {n, 0, 16}] (* _Jean-François Alcover_, Jun 09 2018 *)

%o (PARI) {a(n)=polcoeff(1/(1-sum(k=1,n,x^(k^2))+x*O(x^(n^2))),n^2)}

%o for(n=0,21,print1(a(n),", "))

%o (PARI) {A006456(n)=polcoeff(1/(1-sum(k=1,sqrtint(n+1),x^(k^2))+x*O(x^n)),n)}

%o {a(n)=if(n==0,1,sum(k=1,n,A006456(n^2-k^2)))}

%o for(n=0,21,print1(a(n),", "))

%Y Cf. A006456, A232266.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Apr 05 2013