A224366 - OEIS

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A224366

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

1, 1, 2, 11, 124, 2870, 133462, 12477207, 2344649612, 885591183971, 672331353833716, 1025954712063362545, 3146790000180780110540, 19400015532276248131470280, 240398159948843792847457589388, 5987629866666297470033540284817068, 299759874416459708067727376075503706332

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Equals the row sums of triangle A232266.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..81

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

Cf. A006456, A232266.

Sequence in context: A121337 A269069 A361036 * A342357 A279703 A206401

Adjacent sequences: A224363 A224364 A224365 * A224367 A224368 A224369

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Apr 05 2013

STATUS

approved