A093116 Number of partitions of n^2 into squares not less than n.

1, 1, 1, 1, 2, 2, 2, 1, 2, 5, 4, 4, 5, 9, 15, 23, 24, 13, 20, 32, 55, 84, 113, 185, 303, 545, 167, 298, 435, 716, 1055, 1701, 2584, 4213, 6471, 10218, 15884, 4856, 7376, 11231, 17221, 26054, 39583, 60109, 91622, 138569, 209951, 318368, 483098, 730183

Offset

0,5

Links

Alois P. Heinz, Table of n, a(n) for n = 0..250

Example

n=10: 10^2 = 100 = 64+36 = 36+16+16+16+16 = 25+25+25+25, all other partitions of 100 into squares contain parts < 10, therefore a(10) = 4.

Maple

b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i^2>n, 0, b(n, i+1) +b(n-i^2, i)))
    end:
a:= proc(n) local r; r:= isqrt(n);
      b(n^2, r+`if`(r^2<n, 1, 0))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Apr 15 2013

Mathematica

b[n_, i_] := b[n, i] = If[n==0, 1, If[i^2>n, 0, b[n, i+1] + b[n-i^2, i]]]; a[n_] := With[{r = Sqrt[n]//Floor}, b[n^2, r + If[r^2<n, 1, 0]]]; Table[ a[n], {n, 0, 50}] (* Jean-François Alcover, Oct 26 2015, after Alois P. Heinz *)

Crossrefs

Cf. A093115, A092362, A001156, A037444, A078134.

Cf. A072213, A161408. [Reinhard Zumkeller, Jun 10 2009]

Sequence in context: A353981 A271205 A303841 * A327799 A198199 A366039

Adjacent sequences: A093113 A093114 A093115 * A093117 A093118 A093119

Keyword

nonn

Author

Reinhard Zumkeller, Mar 21 2004

Status

approved