A119915 Number of ternary words of length n and having exactly one run of 0's of odd length.

0, 1, 4, 13, 40, 117, 332, 921, 2512, 6761, 18004, 47525, 124536, 324317, 840092, 2166065, 5562272, 14232273, 36300196, 92321085, 234192584, 592695109, 1496810732, 3772761289, 9492450672, 23844342073, 59804611060, 149787196117

Offset: 0

Author

Emeric Deutsch, May 29 2006

Keywords

• nonn
• easy

Comments

Column 1 of A119914.

Examples

			a(3) = 13 because we have 000, 011, 012, 021, 022, 101, 102, 110, 120, 201, 202, 210 and 220 (for example, 001, 020 do not qualify).
		

Links

• Index entries for linear recurrences with constant coefficients, signature (4,-2,-4,-1).

Crossrefs

Cf. A119914, A193737, A006645.

Programs

Maple

g := z*(1-z^2)/(1-2*z-z^2)^2:
gser := series(g,z=0,34):
seq(coeff(gser,z,n), n=0..30);

Mathematica

LinearRecurrence[ {4, -2, -4, -1}, {0, 1, 4, 13}, 28] (* Peter Luschny, Jan 14 2020 *)

Formula

a(n) = [z^n] z*(1 - z^2)/(1 - 2*z - z^2)^2.

a(n) = A006645(n+1) - A006645(n-1). - R. J. Mathar, Aug 07 2015

From Peter Luschny, Jan 14 2020: (Start)

a(n) = Sum_{k=0..n} A193737(n, k)*k.

Let h(k) = (1 + k)*exp((1 + k)*x)*(1 + x - 1/k)/4 then

a(n) = n!*[x^n](h(sqrt(2)) + h(-sqrt(2))). (End)