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 A287350 Number of independent vertex sets and vertex covers in the n-gear graph. 1
 5, 11, 26, 63, 155, 386, 971, 2463, 6290, 16151, 41651, 107778, 279635, 727031, 1893266, 4936383, 12883115, 33647426, 87928091, 229874703, 601171730, 1572591911, 4114506851, 10766734338, 28177307555, 73748411111, 193034371346, 505287594063, 1322694193115 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Extended to a(1)-a(2) using the formula/recurrence. LINKS Colin Barker, Table of n, a(n) for n = 1..1000 Eric Weisstein's World of Mathematics, Gear Graph Eric Weisstein's World of Mathematics, Independent Vertex Set Eric Weisstein's World of Mathematics, Vertex Cover Index entries for linear recurrences with constant coefficients, signature (5,-7,2). FORMULA a(n) = 2^n + Lucas(2*n). G.f.: x*(5 - 14*x + 6*x^2)/((1 - 2*x)*(1 - 3*x + x^2)). - Ilya Gutkovskiy, May 23 2017 From Colin Barker, Jun 05 2017: (Start) a(n) = 5*a(n-1) - 7*a(n-2) + 2*a(n-3) for n>3. a(n) = 2^(-n)*(4^n + (3-sqrt(5))^n + (3+sqrt(5))^n). (End) MATHEMATICA Table[2^n + LucasL[2 n], {n, 20}] LinearRecurrence[{5, -7, 2}, {5, 11, 26}, 20] CoefficientList[Series[(-5 + 14 x - 6 x^2)/(-1 + 5 x - 7 x^2 + 2 x^3), {x, 0, 20}], x] PROG (Python) from sympy import lucas def a(n): return 2**n + lucas(2*n) # Indranil Ghosh, May 24 2017 (PARI) Vec(x*(5 - 14*x + 6*x^2)/((1 - 2*x)*(1 - 3*x + x^2)) + O(x^30)) \\ Colin Barker, Jun 05 2017 CROSSREFS Cf. A000032, A000079, A005248. Sequence in context: A326161 A038253 A274681 * A294091 A032379 A152535 Adjacent sequences:  A287347 A287348 A287349 * A287351 A287352 A287353 KEYWORD nonn,easy AUTHOR Eric W. Weisstein, May 23 2017 STATUS approved

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Last modified November 28 01:24 EST 2021. Contains 349396 sequences. (Running on oeis4.)