A339445 - OEIS

%I #9 Oct 27 2023 10:01:56

%S 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 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 16,28,21,18,33,30,24,33,33,29,33,37,33,43,45,35,49

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

%H Robert Israel, <a href="/A339445/b339445.txt">Table of n, a(n) for n = 0..2500</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SquareNumber.html">Square Number</a>

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%H <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>

%e                                     [1 1 1]

%e                           [1 4]     [1 1 1]

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

%p g:= proc(n, k, m)

%p   # number of partitions of n into k parts which are squares > m^2

%p    option remember; local r;

%p   if k = 0 then if n = 0 then return 1 else return 0 fi fi;

%p   if n < k*(m+1)^2 then return 0 fi;

%p   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 end proc:

%p f:= proc(n) local k; add(g(n,k^2,0),k=1..floor(sqrt(n))) end proc:

%p f(0):= 1:

%p map(f, [$0..100]); # _Robert Israel_, Oct 26 2023

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

%K nonn

%O 0,5

%A _Ilya Gutkovskiy_, Dec 05 2020