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A080040
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a(n) = 2*a(n-1) + 2*a(n-2) for n > 1; a(0)=2, a(1)=2.
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35
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2, 2, 8, 20, 56, 152, 416, 1136, 3104, 8480, 23168, 63296, 172928, 472448, 1290752, 3526400, 9634304, 26321408, 71911424, 196465664, 536754176, 1466439680, 4006387712, 10945654784, 29904084992, 81699479552, 223207129088, 609813217280, 1666040692736, 4551707820032
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OFFSET
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0,1
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COMMENTS
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The signed version 2, -2, 8, -20, 56, -152, 416, -1136, 3104, -8480, 23168, ... is the Lucas sequence V(-2,-2). - R. J. Mathar, Jan 08 2013
After a(2) equals round((1+sqrt(3))^n) = 1, 3, 7, 20, 56, 152, ... - Jeremy Gardiner, Aug 11 2013
Also the number of independent vertex sets and vertex covers in the n-sunlet graph. - Eric W. Weisstein, Sep 27 2017
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LINKS
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I. Amburg, K. Dasaratha, L. Flapan, T. Garrity, C. Lee, C. Mihailak, N. Neumann-Chun, S. Peluse, and M. Stoffregen, Stern Sequences for a Family of Multidimensional Continued Fractions: TRIP-Stern Sequences, arXiv:1509.05239v1 [math.CO], 2015-2017.
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FORMULA
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G.f.: (2-2*x)/(1-2*x-2*x^2).
a(n) = (1+sqrt(3))^n + (1-sqrt(3))^n.
G.f.: G(0), where G(k) = 1 + 1/(1 - x*(3*k-1)/(x*(3*k+2) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 11 2013
a(n) = [x^n] ( 1 + x + sqrt(1 + 2*x + 3*x^2) )^n for n >= 1. - Peter Bala, Jun 29 2015
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MATHEMATICA
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CoefficientList[Series[(2 - 2 t)/(1 - 2 t - 2 t^2), {t, 0, 30}], t]
With[{c = {2, 2}}, LinearRecurrence[c, c, 20]] (* Harvey P. Dale, Apr 24 2016 *)
Table[(1 - Sqrt[3])^n + (1 + Sqrt[3])^n, {n, 0, 20}] // Expand (* Eric W. Weisstein, Sep 27 2017 *)
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PROG
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(Sage) from sage.combinat.sloane_functions import recur_gen2b; it = recur_gen2b(2, 2, 2, 2, lambda n: 0); [next(it) for i in range(27)] # Zerinvary Lajos, Jul 16 2008
(Sage) [lucas_number2(n, 2, -2) for n in range(0, 27)] # Zerinvary Lajos, Apr 30 2009
(Haskell)
a080040 n = a080040_list !! n
a080040_list =
2 : 2 : map (* 2) (zipWith (+) a080040_list (tail a080040_list))
(Magma) a:=[2, 2]; [n le 2 select a[n] else 2*Self(n-1) + 2*Self(n-2):n in [1..27]]; Marius A. Burtea, Jan 20 2020
(Magma) R<x>:=PowerSeriesRing(Rationals(), 27); Coefficients(R!( (2-2*x)/(1-2*x-2*x^2))); // Marius A. Burtea, Jan 20 2020
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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Mario Catalani (mario.catalani(AT)unito.it), Jan 21 2003
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STATUS
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approved
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