%I #77 Aug 06 2024 02:13:58

%S 3,2,6,17,42,107,273,695,1770,4508,11481,29240,74469,189659,483027,

%T 1230182,3133050,7979309,20321850,51756059,131813277,335704463,

%U 854978262,2177474264,5545631253,14123715032,35970535581,91610417447,233315085507,594211124042,1513347751038,3854221711625,9816002298330

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

%C Also the number of maximal independent vertex sets (and minimal vertex covers) on the 2n-crossed prism graph. - _Eric W. Weisstein_, Jun 15 2017

%C Also the number of irredundant sets in the n-sun graph. - _Eric W. Weisstein_, Aug 07 2017

%C 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

%C 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

%H Robert Israel, <a href="/A276225/b276225.txt">Table of n, a(n) for n = 0..2450</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CrossedPrismGraph.html">Crossed Prism Graph</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IrredundantSet.html">Irredundant Set</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MaximalIndependentVertexSet.html">Maximal Independent Vertex Set</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MinimalVertexCover.html">Minimal Vertex Cover</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SunGraph.html">Sun Graph</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,1).

%F 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)).

%F G.f.: (3 - 4*x - x^2)/(1 - 2*x - x^2 - x^3).

%F 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.

%F a(n) = 3*A077939(n+2) - 4*A077939(n+1) - A077939(n). - _Kai Wang_, Apr 15 2020

%p 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):

%p map(f, [$0..40]); # _Robert Israel_, Aug 29 2016

%t LinearRecurrence[{2, 1, 1}, {3, 2, 6}, 50]

%t CoefficientList[Series[(3 - 4 x - x^2)/(1 - 2 x - x^2 - x^3), {x, 0, 32}], x] (* _Michael De Vlieger_, Aug 25 2016 *)

%t Table[RootSum[-1 - #1 - 2 #1^2 + #1^3 &, #^n &], {n, 20}] (* _Eric W. Weisstein_, Jun 15 2017 *)

%o (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

%o (PARI) Vec((3-4*x-x^2)/(1-2*x-x^2-x^3) + O(x^99)) \\ _Altug Alkan_, Aug 25 2016

%Y Cf. A077939, A276226.

%K nonn,easy

%O 0,1

%A _G. C. Greubel_, Aug 24 2016