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A245968 The edge independence number of the Lucas cube Lambda(n). 1
0, 0, 1, 1, 3, 5, 8, 14, 23, 37, 61, 99, 160, 260, 421, 681, 1103, 1785, 2888, 4674, 7563, 12237, 19801, 32039, 51840, 83880, 135721, 219601, 355323, 574925, 930248, 1505174, 2435423, 3940597, 6376021, 10316619, 16692640, 27009260, 43701901, 70711161, 114413063 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

The Lucas cube Lambda(n) can be defined as the graph whose vertices are the binary strings of length n without either two consecutive 1's or a 1 in the first and in the last position, and in which two vertices are adjacent when their Hamming distance is exactly 1.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

E. Munarini, C. P. Cippo, N. Z. Salvi, On the Lucas cubes, The Fibonacci Quarterly, 39, No. 1, 2001, 12-21.

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

FORMULA

a(n) = floor((L(n)-1)/2), where L(n) = A000032(n) are the Lucas numbers (L(0)=2, L(1)=1, L(n)=L(n-1)+L(n-2) for n>=2).

G.f.: x^2*(x^2+1) / ((x-1)*(x^2+x-1)*(x^2+x+1)). - Colin Barker, Aug 31 2014

a(n) = a(n-1)+a(n-2)+a(n-3)-a(n-4)-a(n-5). - Colin Barker, Aug 31 2014

EXAMPLE

a(3)=1 because Lambda(3) is the star tree on four vertices, having, obviously, vertex independence number equal to 1.

MAPLE

with(combinat): a := proc (n) options operator, arrow: floor((1/2)*fibonacci(n+1)+(1/2)*fibonacci(n-1)-1/2) end proc: seq(a(n), n = 0 .. 40);

MATHEMATICA

CoefficientList[Series[x^2 (x^2 + 1)/((x - 1) (x^2 + x - 1) (x^2 + x + 1)), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 31 2014 *)

PROG

(PARI) concat([0, 0], Vec(x^2*(x^2+1)/((x-1)*(x^2+x-1)*(x^2+x+1)) + O(x^100))) \\ Colin Barker, Aug 31 2014

(MAGMA) [Floor((Lucas(n)-1)/2):n in [0..50]]; // Vincenzo Librandi, Aug 31 2014

CROSSREFS

Cf. A000032.

Sequence in context: A159914 A153251 A229167 * A109022 A023596 A208667

Adjacent sequences:  A245965 A245966 A245967 * A245969 A245970 A245971

KEYWORD

nonn,easy

AUTHOR

Emeric Deutsch, Aug 16 2014

STATUS

approved

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Last modified July 27 01:40 EDT 2021. Contains 346302 sequences. (Running on oeis4.)