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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
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
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
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
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
Sequence in context: A074718 A285457 A007812 * A082561 A268642 A110768
KEYWORD
nonn,easy
AUTHOR
G. C. Greubel, Aug 24 2016
STATUS
approved