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

Author

G. C. Greubel, Aug 24 2016

Keywords

• nonn
• easy

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

Also the number of tilings of a bracelet of length n with two colors of squares and one color of domino and tromino. - Greg Dresden and Arnim Kuchhal, Aug 05 2024

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).

Crossrefs

Cf. A077939, A276226.

Programs

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

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 *)

PARI

Vec((3-4*x-x^2)/(1-2*x-x^2-x^3) + O(x^99)) \\ Altug Alkan, Aug 25 2016

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