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A331983 Number of compositions (ordered partitions) of n into distinct squares > 1.

%I A331983 #12 Feb 24 2021 09:08:52
%S A331983 1,0,0,0,1,0,0,0,0,1,0,0,0,2,0,0,1,0,0,0,2,0,0,0,0,3,0,0,0,8,0,0,0,0,
%T A331983 2,0,1,0,6,0,2,2,0,0,0,8,0,0,0,7,6,0,2,2,24,0,6,0,2,0,0,8,6,0,1,32,0,
%U A331983 0,2,6,6,0,0,2,32,0,0,12,30,0,2
%N A331983 Number of compositions (ordered partitions) of n into distinct squares > 1.
%H A331983 Alois P. Heinz, <a href="/A331983/b331983.txt">Table of n, a(n) for n = 0..20000</a>
%H A331983 <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%H A331983 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%e A331983 a(25) = 3 because we have [25], [16, 9] and [9, 16].
%p A331983 b:= proc(n, i, p) option remember;
%p A331983       `if`(n=0, p!, `if`(i*(i+1)*(2*i+1)/6-1<n, 0,
%p A331983       `if`(i^2>n, 0, b(n-i^2, i-1, p+1))+b(n, i-1, p)))
%p A331983     end:
%p A331983 a:= n-> b(n, isqrt(n), 0):
%p A331983 seq(a(n), n=0..87);  # _Alois P. Heinz_, Feb 03 2020
%t A331983 b[n_, i_, p_] := b[n, i, p] = If[n == 0, p!, If[i(i+1)(2i+1)/6 - 1 < n, 0, If[i^2 > n, 0, b[n - i^2, i - 1, p + 1]] + b[n, i - 1, p]]];
%t A331983 a[n_] := b[n, Floor@Sqrt[n], 0];
%t A331983 a /@ Range[0, 87] (* _Jean-François Alcover_, Nov 26 2020, after _Alois P. Heinz_ *)
%Y A331983 Cf. A000290, A006456, A032020, A032021, A032022, A033461, A078134, A280129, A280542, A331844, A331918.
%K A331983 nonn
%O A331983 0,14
%A A331983 _Ilya Gutkovskiy_, Feb 03 2020