A066258 a(n) = Fibonacci(n)^2 * Fibonacci(n+1).

0, 1, 2, 12, 45, 200, 832, 3549, 14994, 63580, 269225, 1140624, 4831488, 20466953, 86698690, 367262700, 1555747893, 6590256856, 27916771136, 118257348165, 500946152850, 2122041977276, 8989114033297, 38078498156832, 161303106585600, 683290924620625, 2894466804871682

Offset: 0

Author

Len Smiley, Dec 09 2001

Keywords

• nonn
• easy

Comments

From Feryal Alayont, Apr 27 2023: (Start)

a(n) is the number of edge covers of a caterpillar graph with spine P_(3n-2), one pendant attached at vertex n counting from the left end of the spine and another pendant at vertex 2n-1. The caterpillar graph for n=3 is as follows:

* *

| |

*--*--*--*--*--*--*

v

Each pendant edge must be included in an edge cover. Every vertex except v is then incident with at least one edge. Therefore, of the remaining four edges in the spine, only two in the middle have restrictions. At least one of those two has to be in the edge cover to ensure v is incident with one edge in the edge cover, leaving us with 3*2*2 total edge covers. (End)

Links

• Harry J. Smith, Table of n, a(n) for n = 0..200
• Feryal Alayont and Evan Henning, Edge Covers of Caterpillars, Cycles with Pendants, and Spider Graphs, J. Int. Seq. (2023) Vol. 26, Art. 23.9.4.
• David Zeitlin, Generating Functions for Products of Recursive Sequences, Transactions A.M.S., 116, Apr. 1965, p. 304.
• Index entries for linear recurrences with constant coefficients, signature (3,6,-3,-1).

Crossrefs

Cf. A000045, A033887, A056570, A065563, A066259.

First differences of A001655.

Programs

Magma

[Fibonacci(n)^2*Fibonacci(n+1): n in [0..30]]; // G. C. Greubel, Feb 12 2024

Mathematica

#[[1]]^2 #[[2]]&/@Partition[Fibonacci[Range[0,30]],2,1] (* or *) LinearRecurrence[ {3,6,-3,-1},{0,1,2,12},30] (* Harvey P. Dale, Jul 28 2018 *)

PARI

a(n) = { fibonacci(n)^2 * fibonacci(n+1) } \\ Harry J. Smith, Feb 07 2010

SageMath

[fibonacci(n)^2*fibonacci(n+1) for n in range(31)] # G. C. Greubel, Feb 12 2024

Formula

O.g.f.: x*(1-x) / ( (1-4*x-x^2)*(1+x-x^2) ).

a(n) = second term from right in M^(n+1) * [1 0 0 0], where M = the 4 X 4 upper Pascal's triangular matrix [1 3 3 1 / 1 2 1 0 / 1 1 0 0 / 1 0 0 0]. E.g., a(3) = 45 since M^4 * [1 0 0 0] = [125 75 45 27] where 125 = A056570(5), 75 = A066259(4) and 27 = A056570(4). - Gary W. Adamson, Oct 31 2004

a(n) = (1/5)*(Fibonacci(3n+1) - (-1)^n*Fibonacci(n+2)). - Ralf Stephan, Jul 26 2005

a(n) = 3*a(n-1) + 6*a(n-2) - 3*a(n-3) - a(n-4). - Zak Seidov, May 07 2015