A393207 Number of compositions (c1, c2, ...) of n with each c_i <= phi*c_{i-1}, where phi = (1 + sqrt(5))/2 = A001622.

1, 1, 2, 3, 5, 8, 12, 19, 28, 43, 64, 95, 141, 208, 306, 448, 655, 956, 1392, 2024, 2938, 4259, 6170, 8928, 12909, 18651, 26932, 38868, 56067, 80845, 116530, 167916, 241895, 348385, 501650, 722214, 1039590, 1496230, 2153193, 3098292, 4457818, 6413392, 9226210

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..3000

Formula

A000041(n) <= a(n) <= A002843(n) <= A011782(n).

Example

a(5) = 8: 11111, 2111, 221, 23, 311, 32, 41, 5.
a(6) = 12: 111111, 21111, 2211, 222, 231, 3111, 321, 33, 411, 42, 51, 6.
a(7) = 19: 1111111, 211111, 22111, 2221, 223, 2311, 232, 31111, 3211, 322, 331, 34, 4111, 421, 43, 511, 52, 61, 7.

Maple

v:= proc(n) option remember; floor((1+sqrt(5))/2*n) end:
b:= proc(n, i) option remember; `if`(n=0, 1,
      add(b(n-j, min(n-j, v(j))), j=1..i))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..42);

Crossrefs

Cf. A000041, A000201, A001622, A002843, A011782, A224957, A342333.

Sequence in context: A267372 A369696 A355975 * A327421 A124062 A274199

Adjacent sequences: A393204 A393205 A393206 * A393208 A393209 A393210

Keyword

nonn

Author

Alois P. Heinz, Feb 05 2026

Status

approved