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

Author

Alois P. Heinz, Jul 08 2012

Keywords

• nonn

Examples

			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].
		

Links

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

Crossrefs

Column k=1 of A214258.

Cf. A000005, A001622, A072951.

Programs

Maple

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

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