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

0,3

Comments

Column 1 of A119914.

Links

Table of n, a(n) for n=0..27.

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

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)

Example

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).

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 *)

Crossrefs

Cf. A119914, A193737, A006645.

Sequence in context: A342159 A191132 A360606 * A307577 A137744 A027130

Adjacent sequences: A119912 A119913 A119914 * A119916 A119917 A119918

Keyword

nonn,easy

Author

Emeric Deutsch, May 29 2006

Status

approved