A193519 a(n) = (2/3)*Sum_{i=1..n-1} A000129(i)*3^(n-i).

0, 0, 2, 10, 40, 144, 490, 1610, 5168, 16320, 50930, 157546, 484120, 1480080, 4507162, 13683050, 41439200, 125259264, 378051170, 1139641930, 3432176008, 10328516880, 31062778570, 93374780426, 280574458640, 842810055360, 2531053642322, 7599494558890, 22813774416760, 68478238362384

Offset: 0

Author

N. J. A. Sloane, Jul 29 2011

Keywords

• nonn
• easy

Comments

Number of ternary words of length n on {0,1,2} containing the subwords 02 or 20. - Philippe Deléham, Apr 27 2012

Examples

			a(3) = 10 because among the 3^3 = 27 ternary words of length 3 only 10, namely 002, 020, 021, 022, 102, 120, 200, 201, 202, 220 contain the subwords 02 or 20. - _Philippe Deléham_, Apr 27 2012
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Gy. Tasi and F. Mizukami, Quantum algebraic-combinatoric study of the conformational properties of n-alkanes, J. Math. Chemistry, 25, 1999, 55-64 (see Eq. (19)).
• Index entries for linear recurrences with constant coefficients, signature (5,-5,-3).

Crossrefs

Cf. A000129, A002203, A137212.

Programs

Magma

[n le 3 select 2*Floor((n-1)/2) else 5*Self(n-1) -5*Self(n-2) -3*Self(n-3): n in [1..31]]; // G. C. Greubel, Jan 05 2022

Mathematica

Table[(2*3^n - LucasL[n+1, 2])/2, {n, 0, 30}] (* G. C. Greubel, Jan 05 2022 *)

Sage

[(2*3^n - lucas_number2(n+1, 2, -1))/2 for n in (0..30)] # G. C. Greubel, Jan 05 2022

Formula

a(n) = 2*A137212(n).

G.f.: 2*x^2/((1-3*x)*(1-2*x-x^2)). - Philippe Deléham, Apr 27 2012

a(n) = 5*a(n-1) - 5*a(n-2) - 3*a(n-3), a(0) = a(1) = 0, a(2) = 2. - Philippe Deléham, Apr 27 2012

a(n) = (1/2)*(2*3^n - A002203(n+1)). - G. C. Greubel, Jan 05 2022