A331844 - OEIS

%I #15 Oct 29 2020 15:20:53

%S 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 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 24,1,34,150,0,2,12,30,24,0,2,38,150,0,12,78,144,2

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

%H Alois P. Heinz, <a href="/A331844/b331844.txt">Table of n, a(n) for n = 0..20000</a>

%H <a href="/index/Com#comp">Index entries for sequences related to compositions</a>

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

%e 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 b:= proc(n, i, p) option remember;

%p       `if`(i*(i+1)*(2*i+1)/6<n, 0, `if`(n=0, p!,

%p       `if`(i^2>n, 0, b(n-i^2, i-1, p+1))+b(n, i-1, p)))

%p     end:

%p a:= n-> b(n, isqrt(n), 0):

%p seq(a(n), n=0..82);  # _Alois P. Heinz_, Jan 30 2020

%t 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 a[n_] := b[n, Sqrt[n] // Floor, 0];

%t a /@ Range[0, 82] (* _Jean-François Alcover_, Oct 29 2020, after _Alois P. Heinz_ *)

%Y Cf. A000290, A006456, A032020, A032021, A032022, A033461, A218396, A219107, A331843, A331845, A331846, A331847.

%K nonn

%O 0,6

%A _Ilya Gutkovskiy_, Jan 29 2020