A214259 Number of compositions of n where the difference between largest and smallest parts equals one.

0, 0, 2, 3, 9, 11, 25, 35, 60, 96, 157, 241, 401, 637, 1019, 1639, 2651, 4258, 6870, 11075, 17891, 28895, 46678, 75412, 121915, 197109, 318724, 515414, 833590, 1348301, 2181020, 3528138, 5707564, 9233625, 14938477, 24168522, 39102322, 63264680, 102358836

Offset

1,3

Links

Alois P. Heinz, Table of n, a(n) for n = 1..4785

Formula

a(n) = A072951(n) - A000005(n).

a(n) ~ phi^(n+1) / sqrt(5), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jan 07 2019

Example

a(3) = 2: [2,1], [1,2].
a(4) = 3: [2,1,1], [1,2,1], [1,1,2].
a(5) = 9: [3,2], [2,3], [2,2,1], [2,1,2], [1,2,2], [2,1,1,1], [1,2,1,1], [1,1,2,1], [1,1,1,2].
a(6) = 11: [2,2,1,1], [2,1,2,1], [2,1,1,2], [1,2,2,1], [1,2,1,2], [1,1,2,2], [2,1,1,1,1], [1,2,1,1,1], [1,1,2,1,1], [1,1,1,2,1], [1,1,1,1,2].

Maple

with(numtheory):
a:= n-> add(binomial(t, n mod t), t=1..n) -tau(n):
seq(a(n), n=1..50);

Crossrefs

Column k=1 of A214258.

Cf. A000005, A001622, A072951.

Sequence in context: A098016 A372073 A089645 * A287680 A242680 A275767

Adjacent sequences: A214256 A214257 A214258 * A214260 A214261 A214262

Keyword

nonn

Author

Alois P. Heinz, Jul 08 2012

Status

approved