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A331843
Number of compositions (ordered partitions) of n into distinct triangular numbers.
14
1, 1, 0, 1, 2, 0, 1, 2, 0, 2, 7, 2, 0, 2, 6, 1, 4, 6, 2, 12, 24, 3, 8, 0, 8, 32, 6, 2, 13, 26, 6, 34, 36, 0, 32, 150, 3, 20, 50, 14, 54, 126, 32, 32, 12, 55, 160, 78, 122, 44, 174, 4, 72, 294, 36, 201, 896, 128, 62, 180, 176, 164, 198, 852, 110, 320, 159, 212, 414
OFFSET
0,5
EXAMPLE
a(10) = 7 because we have [10], [6, 3, 1], [6, 1, 3], [3, 6, 1], [3, 1, 6], [1, 6, 3] and [1, 3, 6].
MAPLE
h:= proc(n) option remember; `if`(n<1, 0,
`if`(issqr(8*n+1), 1+h(n-1), h(n-1)))
end:
b:= proc(n, i, p) option remember; (t->
`if`(t*(i+2)/3<n, 0, `if`(n=0, p!, b(n, i-1, p)+
`if`(t>n, 0, b(n-t, i-1, p+1)))))((i*(i+1)/2))
end:
a:= n-> b(n, h(n), 0):
seq(a(n), n=0..73); # Alois P. Heinz, Jan 31 2020
MATHEMATICA
h[n_] := h[n] = If[n<1, 0, If[IntegerQ @ Sqrt[8n+1], 1 + h[n-1], h[n-1]]];
b[n_, i_, p_] := b[n, i, p] = Function[t, If[t (i + 2)/3 < n, 0, If[n == 0, p!, b[n, i-1, p] + If[t>n, 0, b[n - t, i - 1, p + 1]]]]][(i(i + 1)/2)];
a[n_] := b[n, h[n], 0];
a /@ Range[0, 73] (* Jean-François Alcover, Apr 27 2020, after Alois P. Heinz *)
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Jan 29 2020
STATUS
approved