A168445 Number of compositions a(1),...,a(k) of n, for some k, such that a(i+1) <= a(i) + 1 for 1 <= i < k and a(1) <= a(k) + 1.

1, 2, 4, 6, 11, 18, 31, 52, 91, 155, 268, 464, 802, 1390, 2411, 4178, 7249, 12578, 21823, 37870, 65724, 114061, 197960, 343578, 596317, 1034983, 1796359, 3117837, 5411478, 9392460, 16302081, 28294850, 49110242, 85238716, 147945552, 256783448, 445689300

Offset: 1

Author

Vladeta Jovovic, Nov 25 2009

Keywords

• nonn

References

• Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 381.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..400

Crossrefs

Cf. A003116, A005169.

Programs

Maple

b:= proc(n,r,f) option remember; `if`(n=0, `if`(f-1<=r, 1, 0),
      add(b(n-i, i, f), i=1..min(r+1, n)))
    end:
a:= n-> add(b(n-i, i, i), i=1..n):
seq(a(n), n=1..40);  # Alois P. Heinz, Dec 15 2009

Mathematica

b[n_, r_, f_] := b[n, r, f] = If[n == 0, If[f - 1 <= r, 1, 0], Sum[b[n - i, i, f], {i, 1, Min [r + 1, n]}]];
a[n_] := Sum[b[n - i, i, i], {i, 1, n}];
Array[a, 40] (* Jean-François Alcover, Nov 10 2020, after Alois P. Heinz *)

Formula

a(n) ~ c / r^n, where r = A347901 = 0.576148769142756602297868573719938782354724663118974... is the lowest root of the equation Sum_{k>=0} (-1)^k * r^(k^2) / QPochhammer(r, r, k) = 0 and c = 0.6149126319329581124890112676009720339906790088212712130894... - Vaclav Kotesovec, May 01 2014, updated Sep 09 2020

Extensions

More terms from Alois P. Heinz, Dec 15 2009