A093116 - OEIS

%I #11 Oct 26 2015 07:51:00

%S 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,

%T 167,298,435,716,1055,1701,2584,4213,6471,10218,15884,4856,7376,11231,

%U 17221,26054,39583,60109,91622,138569,209951,318368,483098,730183

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

%H Alois P. Heinz, <a href="/A093116/b093116.txt">Table of n, a(n) for n = 0..250</a>

%e 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.

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

%p      `if`(i^2>n, 0, b(n, i+1) +b(n-i^2, i)))

%p     end:

%p a:= proc(n) local r; r:= isqrt(n);

%p       b(n^2, r+`if`(r^2<n, 1, 0))

%p     end:

%p seq(a(n), n=0..50);  # _Alois P. Heinz_, Apr 15 2013

%t 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_ *)

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

%Y Cf. A072213, A161408. [_Reinhard Zumkeller_, Jun 10 2009]

%K nonn

%O 0,5

%A _Reinhard Zumkeller_, Mar 21 2004