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A331844 Number of compositions (ordered partitions) of n into distinct squares.

%I A331844 #15 Oct 29 2020 15:20:53
%S A331844 1,1,0,0,1,2,0,0,0,1,2,0,0,2,6,0,1,2,0,0,2,6,0,0,0,3,8,0,0,8,30,0,0,0,
%T A331844 2,6,1,2,6,24,2,8,6,0,0,8,30,0,0,7,32,24,2,8,30,120,6,24,2,6,0,8,36,
%U A331844 24,1,34,150,0,2,12,30,24,0,2,38,150,0,12,78,144,2
%N A331844 Number of compositions (ordered partitions) of n into distinct squares.
%H A331844 Alois P. Heinz, <a href="/A331844/b331844.txt">Table of n, a(n) for n = 0..20000</a>
%H A331844 <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%H A331844 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%e A331844 a(14) = 6 because we have [9,4,1], [9,1,4], [4,9,1], [4,1,9], [1,9,4] and [1,4,9].
%p A331844 b:= proc(n, i, p) option remember;
%p A331844       `if`(i*(i+1)*(2*i+1)/6<n, 0, `if`(n=0, p!,
%p A331844       `if`(i^2>n, 0, b(n-i^2, i-1, p+1))+b(n, i-1, p)))
%p A331844     end:
%p A331844 a:= n-> b(n, isqrt(n), 0):
%p A331844 seq(a(n), n=0..82);  # _Alois P. Heinz_, Jan 30 2020
%t A331844 b[n_, i_, p_] := b[n, i, p] = If[i(i+1)(2i+1)/6 < n, 0, If[n == 0, p!, If[i^2 > n, 0, b[n - i^2, i - 1, p + 1]] + b[n, i - 1, p]]];
%t A331844 a[n_] := b[n, Sqrt[n] // Floor, 0];
%t A331844 a /@ Range[0, 82] (* _Jean-François Alcover_, Oct 29 2020, after _Alois P. Heinz_ *)
%Y A331844 Cf. A000290, A006456, A032020, A032021, A032022, A033461, A218396, A219107, A331843, A331845, A331846, A331847.
%K A331844 nonn
%O A331844 0,6
%A A331844 _Ilya Gutkovskiy_, Jan 29 2020