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A224366
Number of compositions of n^2 into sums of squares.
13
1, 1, 2, 11, 124, 2870, 133462, 12477207, 2344649612, 885591183971, 672331353833716, 1025954712063362545, 3146790000180780110540, 19400015532276248131470280, 240398159948843792847457589388, 5987629866666297470033540284817068, 299759874416459708067727376075503706332
OFFSET
0,3
COMMENTS
Equals the row sums of triangle A232266.
LINKS
FORMULA
a(n) = [x^(n^2)] 1/(1 - Sum_{k>=1} x^(k^2)).
a(n) = A006456(n^2).
a(n) = Sum_{k=1..n} A006456(n^2-k^2) for n>=1 with a(0)=1.
EXAMPLE
Illustrate a(n) = Sum_{k=1..n} A006456(n^2-k^2):
a(1) = 1 = 1;
a(2) = 2 = 1 + 1;
a(3) = 11 = 7 + 3 + 1;
a(4) = 124 = 88 + 30 + 5 + 1;
a(5) = 2870 = 2024 + 710 + 124 + 11 + 1;
a(6) = 133462 = 94137 + 33033 + 5767 + 502 + 22 + 1;
a(7) = 12477207 = 8800750 + 3088365 + 539192 + 46832 + 2024 + 43 + 1; ...
MAPLE
b:= proc(n) option remember; local i; if n=0 then 1
else 0; for i while i^2<=n do %+b(n-i^2) od fi
end:
a:= n-> b(n^2):
seq(a(n), n=0..17); # Alois P. Heinz, Aug 12 2017
MATHEMATICA
b[0] = 1; b[n_] := b[n] = Sum[b[n-k], {k, Select[Range[n], IntegerQ[ Sqrt[#]]&]}];
a[n_] := b[n^2];
Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Jun 09 2018 *)
PROG
(PARI) {a(n)=polcoeff(1/(1-sum(k=1, n, x^(k^2))+x*O(x^(n^2))), n^2)}
for(n=0, 21, print1(a(n), ", "))
(PARI) {A006456(n)=polcoeff(1/(1-sum(k=1, sqrtint(n+1), x^(k^2))+x*O(x^n)), n)}
{a(n)=if(n==0, 1, sum(k=1, n, A006456(n^2-k^2)))}
for(n=0, 21, print1(a(n), ", "))
CROSSREFS
Sequence in context: A121337 A269069 A361036 * A342357 A279703 A206401
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 05 2013
STATUS
approved