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A357354 Number of partitions of n into distinct positive squares such that the number of parts is a square.

%I A357354 #21 Aug 27 2025 21:29:31
%S A357354 1,1,0,0,1,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,
%T A357354 0,0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,1,1,0,0,1,0,0,1,0,0,0,0,1,1,1,1,1,0,
%U A357354 0,0,1,1,0,0,1,1,0,0,3,1,0,2,0,0,1,1,1
%N A357354 Number of partitions of n into distinct positive squares such that the number of parts is a square.
%H A357354 Alois P. Heinz, <a href="/A357354/b357354.txt">Table of n, a(n) for n = 0..20000</a>
%e A357354 a(30) = 1 because we have [16,9,4,1].
%e A357354 a(78) = 3: [36,25,16,1], [49,16,9,4], [64,9,4,1].
%p A357354 b:= proc(n, i, t) option remember; `if`(n=0,
%p A357354      `if`(issqr(t), 1, 0), `if`(n>i*(i+1)*(2*i+1)/6, 0,
%p A357354      `if`(i^2>n, 0, b(n-i^2, i-1, t+1))+b(n, i-1, t)))
%p A357354     end:
%p A357354 a:= n-> b(n, isqrt(n), 0):
%p A357354 seq(a(n), n=0..100);  # _Alois P. Heinz_, Sep 25 2022
%t A357354 b[n_, i_, t_] := b[n, i, t] = If[n == 0,
%t A357354    If[IntegerQ @ Sqrt[t], 1, 0], If[n > i*(i+1)*(2*i+1)/6, 0,
%t A357354    If[i^2 > n, 0, b[n-i^2, i-1, t+1]] + b[n, i-1, t]]];
%t A357354 a[n_] := b[n, Floor @ Sqrt[n], 0];
%t A357354 Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Apr 24 2025, after _Alois P. Heinz_ *)
%Y A357354 Cf. A000290, A033461, A045450, A089333, A357352.
%K A357354 nonn,look,changed
%O A357354 0,79
%A A357354 _Ilya Gutkovskiy_, Sep 25 2022