cp's OEIS Frontend

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

%I A066258 #72 Dec 27 2024 10:47:56
%S A066258 0,1,2,12,45,200,832,3549,14994,63580,269225,1140624,4831488,20466953,
%T A066258 86698690,367262700,1555747893,6590256856,27916771136,118257348165,
%U A066258 500946152850,2122041977276,8989114033297,38078498156832,161303106585600,683290924620625,2894466804871682
%N A066258 a(n) = Fibonacci(n)^2 * Fibonacci(n+1).
%C A066258 From _Feryal Alayont_, Apr 27 2023: (Start)
%C A066258 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:
%C A066258         *     *
%C A066258         |     |
%C A066258   *--*--*--*--*--*--*
%C A066258            v
%C A066258 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)
%H A066258 Harry J. Smith, <a href="/A066258/b066258.txt">Table of n, a(n) for n = 0..200</a>
%H A066258 Feryal Alayont and Evan Henning, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Alayont/ala4.html">Edge Covers of Caterpillars, Cycles with Pendants, and Spider Graphs</a>, J. Int. Seq. (2023) Vol. 26, Art. 23.9.4.
%H A066258 David Zeitlin, <a href="http://dx.doi.org/10.1090/S0002-9947-1965-0185301-0">Generating Functions for Products of Recursive Sequences</a>, Transactions A.M.S., 116, Apr. 1965, p. 304.
%H A066258 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,6,-3,-1).
%F A066258 O.g.f.: x*(1-x) / ( (1-4*x-x^2)*(1+x-x^2) ).
%F A066258 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
%F A066258 a(n) = (1/5)*(Fibonacci(3n+1) - (-1)^n*Fibonacci(n+2)). - _Ralf Stephan_, Jul 26 2005
%F A066258 a(n) = 3*a(n-1) + 6*a(n-2) - 3*a(n-3) - a(n-4). - _Zak Seidov_, May 07 2015
%t A066258 #[[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 *)
%o A066258 (PARI) a(n) = { fibonacci(n)^2 * fibonacci(n+1) } \\ _Harry J. Smith_, Feb 07 2010
%o A066258 (Magma) [Fibonacci(n)^2*Fibonacci(n+1): n in [0..30]]; // _G. C. Greubel_, Feb 12 2024
%o A066258 (SageMath) [fibonacci(n)^2*fibonacci(n+1) for n in range(31)] # _G. C. Greubel_, Feb 12 2024
%Y A066258 Cf. A000045, A033887, A056570, A065563, A066259.
%Y A066258 First differences of A001655.
%K A066258 nonn,easy
%O A066258 0,3
%A A066258 _Len Smiley_, Dec 09 2001