A130872 Number of ways to write 8n - 1 as a sum of 7 (not necessarily distinct) odd squares.

1, 1, 1, 2, 2, 2, 4, 4, 3, 5, 5, 5, 7, 7, 6, 9, 10, 8, 11, 11, 10, 15, 14, 12, 16, 17, 15, 19, 19, 17, 22, 24, 20, 25, 27, 23, 30, 29, 26, 32, 34, 30, 36, 38, 33, 40, 43, 37, 44, 47, 41, 50, 52, 45, 53, 55, 50, 58, 62, 54, 63, 70, 59, 68, 71, 64, 75, 79, 70, 79, 85, 77, 85, 89, 81

Offset

1,4

Comments

Since all odd squares are congruent to 1 mod 8, it is not possible to express any number congruent to 7 mod 8 as a sum of fewer than 7 odd squares.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Example

a(2) = 1 because 15 = 9 + 1 + 1 + 1 + 1 + 1 + 1 is the only such representation.
a(4) = 2 because 31 = 25 + 1 + 1 + 1 + 1 + 1 + 1 = 9 + 9 + 9 + 1 + 1 + 1 + 1.

Maple

A130872recur := proc(n, jmin, N) local a, s ; if N =1 then if n mod 2 = 1 and issqr(n) and n>=jmin^2 then RETURN(1) ; else RETURN(0) ; fi ; else a := 0 ; for s from 2*floor(jmin/2)+1 to floor(sqrt(n)) by 2 do a := a+A130872recur(n-s^2, s, N-1) ; od ; RETURN(a) ; fi ; end: A130872 := proc(n) option remember: A130872recur(8*n-1, 1, 7) ; end: seq(A130872(n), n=1..100); # R. J. Mathar, Aug 02 2007
# second Maple program:
b:= proc(n, i) option remember; convert(series(`if`(n=0, 1, `if`(i<1, 0,
      expand(add(b(n-i^2*j, i-2)*x^j, j=0..min(7, n/i^2))))), x, 8), polynom)
    end:
a:= n-> coeff(b(8*n-1, (t-> `if`(t::odd, t, t-1))(isqrt(8*n-1))), x, 7):
seq(a(n), n=1..100);  # Alois P. Heinz, Jan 27 2024

Mathematica

a[n_] := IntegerPartitions[8 n - 1, {7}, Select[Range[1, 8 n - 1, 2], IntegerQ[Sqrt[#]]&]] // Length;
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jan 27 2024 *)

Crossrefs

Sequence in context: A216326 A101909 A215676 * A087627 A195051 A219654

Adjacent sequences: A130869 A130870 A130871 * A130873 A130874 A130875

Keyword

nonn

Author

Joel B. Lewis, Jul 24 2007

Extensions

More terms from R. J. Mathar, Aug 02 2007

Status

approved