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

%I #12 Feb 24 2021 09:08:52

%S 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 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 0,2,6,6,0,0,2,32,0,0,12,30,0,2

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

%H Alois P. Heinz, <a href="/A331983/b331983.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(25) = 3 because we have [25], [16, 9] and [9, 16].

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

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

%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..87); # _Alois P. Heinz_, Feb 03 2020

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

%t a /@ Range[0, 87] (* _Jean-François Alcover_, Nov 26 2020, after _Alois P. Heinz_ *)

%Y Cf. A000290, A006456, A032020, A032021, A032022, A033461, A078134, A280129, A280542, A331844, A331918.

%K nonn

%O 0,14

%A _Ilya Gutkovskiy_, Feb 03 2020