A095977 Expansion of g.f. 2*x / ((1+x)^2*(1-2*x)^2).

2, 4, 14, 32, 82, 188, 438, 984, 2202, 4852, 10622, 23056, 49762, 106796, 228166, 485448, 1029162, 2174820, 4582670, 9631360, 20194802, 42253724, 88235734, 183927992, 382769082, 795364308, 1650380958, 3420066544, 7078742402, 14634703372, 30223843942, 62356562216

Offset: 1

Author

Ralf Stephan, Jul 16 2004

Keywords

• nonn
• easy

Comments

Number of 2 X 2 tiles in all tilings of a 3 X (n+1) rectangle with 1 X 1 and 2 X 2 square tiles. - Emeric Deutsch, Feb 18 2007

The terms of this sequence have a primitive divisor for all terms beyond the 4th if and only if n is not of the form 4k+2, for some nonnegative integer k. - Anthony Flatters (Anthony.Flatters(AT)uea.ac.uk), Aug 17 2007

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Luca Ferrari and Emanuele Munarini, Enumeration of edges in some lattices of paths, arXiv preprint arXiv:1203.6792 [math.CO], 2012 and J. Int. Seq. 17 (2014) #14.1.5
• A. Flatters, Prime divisors of some Lehmer-Pierce sequences, arXiv:0708.2190 [math.NT], 2007.
• R. P. Grimaldi, Tilings, Compositions, and Generalizations, J. Int. Seq. 13 (2010), 10.6.5, page 7.
• Luka Podrug, Horadam cubes, arXiv:2410.03193 [math.CO], 2024. See p. 11.
• Helmut Prodinger, On binary representations of integers with digits -1,0,1, Integers 0 (2000), #A08.
• Index entries for linear recurrences with constant coefficients, signature (2,3,-4,-4).

Crossrefs

Cf. A128099, A073371.

Programs

Maple

a:=n->n/9*2^(n+2)+1/27*2^(n+3)-2*n/9*(-1)^n-8/27*(-1)^n: seq(a(n),n=1..30); # Emeric Deutsch, Feb 18 2007

Mathematica

Table[(1/27)*((3*n + 2)*2^(n + 2) - (6*n + 8)*(-1)^n) , {n,1,50}] (* G. C. Greubel, Dec 28 2016 *)

PARI

Vec(2*x / ((1+x)^2 * (1-2*x)^2) + O(x^50)) \\ Michel Marcus, Nov 07 2015

Formula

a(n) = (1/27)*((3*n + 2)*2^(n + 2) - (6*n + 8)*(-1)^n).

a(n) = 2 * A073371(n-1).

a(n) = Sum_{k=0..floor((n+1)/2)} k*2^k*binomial(n+1-k,k). - Emeric Deutsch, Feb 18 2007

E.g.f.: 2*(cosh(x/2) + sinh(x/2))*(15*x*cosh(3*x/2) + (8 + 9*x)*sinh(3*x/2))/27. - Stefano Spezia, Oct 12 2024