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 A276225 a(n+3) = 2*a(n+2) + a(n+1) + a(n) with a(0)=3, a(1)=2, a(2)=6. 4
 3, 2, 6, 17, 42, 107, 273, 695, 1770, 4508, 11481, 29240, 74469, 189659, 483027, 1230182, 3133050, 7979309, 20321850, 51756059, 131813277, 335704463, 854978262, 2177474264, 5545631253, 14123715032, 35970535581, 91610417447, 233315085507, 594211124042, 1513347751038, 3854221711625, 9816002298330 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS Also the number of maximal independent vertex sets (and minimal vertex covers) on the 2n-crossed prism graph. - Eric W. Weisstein, Jun 15 2017 Also the number of irredundant sets in the n-sun graph. - Eric W. Weisstein, Aug 07 2017 Let {x,y,z} be the simple roots of P(x) = x^3 + u*x^2 + v*x + w. For n>=0, let p(n) = x^n/((x-y)(x-z)) + y^n/((y-x)(y-z)) + z^n/((z-x)(z-y)), q(n) =  x^n + y^n + z^n. Then for n >= 0, q(n) = 3*p(n+2) + 2*u*p(n+1) + v*p(n). In this case, P(x) = x^3 - 2*x^2 - x - 1, q(n) = a(n), p(n) = A077939(n). - Kai Wang, Apr 15 2020 LINKS Robert Israel, Table of n, a(n) for n = 0..2450 Eric Weisstein's World of Mathematics, Crossed Prism Graph Eric Weisstein's World of Mathematics, Irredundant Set Eric Weisstein's World of Mathematics, Maximal Independent Vertex Set Eric Weisstein's World of Mathematics, Minimal Vertex Cover Eric Weisstein's World of Mathematics, Sun Graph Index entries for linear recurrences with constant coefficients, signature (2,1,1). FORMULA Let p = (4*(61 + 9*sqrt(29)))^(1/3), q = (4*(61 - 9*sqrt(29)))^(1/3), and x = (1/6)*(4 + p + q) then x^n = (1/6)*(2*a(n) + A276226(n)*(p + q) + A077939(n-1)*(p^2 + q^2)). G.f.: (3 - 4*x - x^2)/(1 - 2*x - x^2 - x^3). a(n) = b^n + c^n + d^n, where (b, c, d) are the three roots of the cubic equation x^3 = 2*x^2 + x + 1. a(n) = 3*A077939(n+2) - 4*A077939(n+1) - A077939(n). - Kai Wang, Apr 15 2020 MAPLE f:= gfun:-rectoproc({a(n+3) = 2*a(n+2) + a(n+1) + a(n), a(0)=3, a(1)=2, a(2)=6}, a(n), remember): map(f, [\$0..40]); # Robert Israel, Aug 29 2016 MATHEMATICA LinearRecurrence[{2, 1, 1}, {3, 2, 6}, 50] CoefficientList[Series[(3 - 4 x - x^2)/(1 - 2 x - x^2 - x^3), {x, 0, 32}], x] (* Michael De Vlieger, Aug 25 2016 *) Table[RootSum[-1 - #1 - 2 #1^2 + #1^3 &, #^n &], {n, 20}] (* Eric W. Weisstein, Jun 15 2017 *) PROG (MAGMA) I:=[3, 2, 6]; [n le 3 select I[n] else 2*Self(n-1)+ Self(n-2)+Self(n-3): n in [1..40]]; // Vincenzo Librandi, Aug 25 2016 (PARI) Vec((3-4*x-x^2)/(1-2*x-x^2-x^3) + O(x^99)) \\ Altug Alkan, Aug 25 2016 CROSSREFS Cf. A077939, A276226. Sequence in context: A074718 A285457 A007812 * A082561 A268642 A110768 Adjacent sequences:  A276222 A276223 A276224 * A276226 A276227 A276228 KEYWORD nonn,easy AUTHOR G. C. Greubel, Aug 24 2016 STATUS approved

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Last modified August 5 07:02 EDT 2021. Contains 346458 sequences. (Running on oeis4.)