A276225 a(n+3) = 2*a(n+2) + a(n+1) + a(n) with a(0)=3, a(1)=2, a(2)=6.

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

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