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

%I A339445 #10 Feb 16 2025 08:34:01
%S A339445 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,
%T A339445 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,
%U A339445 16,28,21,18,33,30,24,33,33,29,33,37,33,43,45,35,49
%N A339445 Number of partitions of n into squares such that the number of parts is a square.
%H A339445 Robert Israel, <a href="/A339445/b339445.txt">Table of n, a(n) for n = 0..2500</a>
%H A339445 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SquareNumber.html">Square Number</a>
%H A339445 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%H A339445 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%e A339445                                     [1 1 1]
%e A339445                           [1 4]     [1 1 1]
%e A339445 a(23) = 2 because we have [9 9] and [4 4 9].
%p A339445 g:= proc(n, k, m)
%p A339445   # number of partitions of n into k parts which are squares > m^2
%p A339445    option remember; local r;
%p A339445   if k = 0 then if n = 0 then return 1 else return 0 fi fi;
%p A339445   if n < k*(m+1)^2 then return 0 fi;
%p A339445   add(procname(n-r*(m+1)^2, k-r, m+1), r =max(0, ceil((k*(m+2)^2-n)/(2*m+3))) .. k)
%p A339445 end proc:
%p A339445 f:= proc(n) local k; add(g(n,k^2,0),k=1..floor(sqrt(n))) end proc:
%p A339445 f(0):= 1:
%p A339445 map(f, [$0..100]); # _Robert Israel_, Oct 26 2023
%Y A339445 Cf. A000290, A001156, A085755, A089333, A103198, A323522, A339444.
%K A339445 nonn
%O A339445 0,5
%A A339445 _Ilya Gutkovskiy_, Dec 05 2020