A339445 Number of partitions of n into squares such that the number of parts is a square.

1, 1, 0, 0, 2, 0, 0, 1, 0, 2, 1, 0, 2, 1, 0, 2, 3, 1, 2, 2, 2, 2, 2, 2, 3, 5, 2, 4, 6, 1, 4, 6, 3, 7, 6, 4, 10, 6, 4, 10, 9, 6, 11, 10, 8, 10, 10, 11, 14, 16, 11, 15, 19, 10, 17, 22, 13, 24, 23, 16, 28, 21, 18, 33, 30, 24, 33, 33, 29, 33, 37, 33, 43, 45, 35, 49

Offset: 0

Author

Ilya Gutkovskiy, Dec 05 2020

Keywords

• nonn

Examples

			                                    [1 1 1]
                          [1 4]     [1 1 1]
a(23) = 2 because we have [9 9] and [4 4 9].
		

Links

• Robert Israel, Table of n, a(n) for n = 0..2500
• Eric Weisstein's World of Mathematics, Square Number
• Index entries for sequences related to partitions
• Index entries for sequences related to sums of squares

Crossrefs

Cf. A000290, A001156, A085755, A089333, A103198, A323522, A339444.

Programs

Maple

g:= proc(n, k, m)
  # number of partitions of n into k parts which are squares > m^2
   option remember; local r;
  if k = 0 then if n = 0 then return 1 else return 0 fi fi;
  if n < k*(m+1)^2 then return 0 fi;
  add(procname(n-r*(m+1)^2, k-r, m+1), r =max(0, ceil((k*(m+2)^2-n)/(2*m+3))) .. k)
end proc:
f:= proc(n) local k; add(g(n,k^2,0),k=1..floor(sqrt(n))) end proc:
f(0):= 1:
map(f, [$0..100]); # Robert Israel, Oct 26 2023