A006497 a(n) = 3*a(n-1) + a(n-2) with a(0) = 2, a(1) = 3. (Formerly M0910)

2, 3, 11, 36, 119, 393, 1298, 4287, 14159, 46764, 154451, 510117, 1684802, 5564523, 18378371, 60699636, 200477279, 662131473, 2186871698, 7222746567, 23855111399, 78788080764, 260219353691, 859446141837, 2838557779202

Offset

0,1

Comments

For more information about this type of recurrence follow the Khovanova link and see A086902 and A054413. - Johannes W. Meijer, Jun 12 2010

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

P. Bhadouria, D. Jhala, and B. Singh, Binomial Transforms of the k-Lucas Sequences and its [sic] Properties, Journal of Mathematics and Computer Science (JMCS), Volume 8, Issue 1, Pages 81-92, Sequence L_{3,n}.

A. F. Horadam, Generating identities for generalized Fibonacci and Lucas triples, Fib. Quart., 15 (1977), 289-292.

Haruo Hosoya, What Can Mathematical Chemistry Contribute to the Development of Mathematics?, HYLE--International Journal for Philosophy of Chemistry, Vol. 19, No.1 (2013), pp. 87-105.

Tanya Khovanova, Recursive Sequences

Pablo Lam-Estrada, Myriam Rosalía Maldonado-Ramírez, José Luis López-Bonilla, and Fausto Jarquín-Zárate, The sequences of Fibonacci and Lucas for each real quadratic fields Q(Sqrt(d)), arXiv:1904.13002 [math.NT], 2019.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (3,1).

Formula

G.f.: (2-3*x)/(1-3*x-x^2). - Simon Plouffe in his 1992 dissertation

From Gary W. Adamson, Jun 15 2003: (Start)

a(n) = ((3 + sqrt(13))/2)^n + ((3 - sqrt(13))/2)^n. See bronze mean (A098316).

A006190(n-2) + A006190(n) = a(n-1).

a(n)^2 - 13*A006190(n)^2 = 4(-1)^n. (End)

From Paul Barry, Nov 15 2003: (Start)

E.g.f.: 2*exp(3*x/2)*cosh(sqrt(13)*x/2).

a(n) = 2^(1-n)*Sum_{k=0..floor(n/2)} C(n, 2*k)* (13)^k * 3^(n-2*k).

a(n) = 2*T(n, 3i/2)*(-i)^n with T(n, x) Chebyshev's polynomials of the first kind (see A053120) and i^2=-1. (End)

From Hieronymus Fischer, Jan 02 2009: (Start)

fract(((3+sqrt(13))/2)^n)) = (1/2)*(1+(-1)^n) - (-1)^n*((3+sqrt(13))/2)^(-n) = (1/2)*(1+(-1)^n) - ((3-sqrt(13))/2)^n.

See A001622 for a general formula concerning the fractional parts of powers of numbers x>1, which satisfy x-x^(-1)=floor(x).

a(n) = round(((3+sqrt(13))/2)^n) for n > 0. (End)

From Johannes W. Meijer, Jun 12 2010: (Start)

a(2n+1) = 3*A097783(n), a(2n) = A057076(n).

a(3n+1) = A041018(5n), a(3n+2) = A041018(5n+3) and a(3n+3) = 2*A041018(5n+4).

Limit_{k -> infinity} a(n+k)/a(k) = (a(n) + A006190(n)*sqrt(13))/2.

Limit_{n -> infinity} a(n)/A006190(n) = sqrt(13).

(End)

a(n) = sqrt(13*(A006190(n))^2 + 4*(-1)^n). - Vladimir Shevelev, Mar 13 2013

G.f.: G(0), where G(k) = 1 + 1/(1 - (x*(13*k-9))/((x*(13*k+4)) - 6/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 15 2013

a(n) = [x^n] ( (1 + 3*x + sqrt(1 + 6*x + 13*x^2))/2 )^n for n >= 1. - Peter Bala, Jun 23 2015

a(n) = Lucas(n,3), Lucas polynomials, L(n,x), evaluated at x=3. - G. C. Greubel, Jun 06 2019

a(n) = 2 * Sum_{k=0..n-2} A168561(n-2,k)*3^k + 3 * Sum_{k=0..n-1} A168561(n-1,k)*3^k, n>0. - R. J. Mathar, Feb 14 2024

a(n) = 2*A006190(n+1) - 3*A006190(n). - R. J. Mathar, Feb 14 2024

Maple

a:= n-> (<<0|1>, <1|3>>^n. <<2, 3>>)[1, 1]:
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 26 2018

Mathematica

Table[LucasL[n, 3], {n, 0, 30}] (* Zerinvary Lajos, Jul 09 2009 *)
LucasL[Range[0, 30], 3] (* Eric W. Weisstein, Apr 17 2018 *)
LinearRecurrence[{3, 1}, {2, 3}, 30] (* Harvey P. Dale, Feb 17 2020 *)

Prog

(Sage) [lucas_number2(n, 3, -1) for n in range(0, 30)] # Zerinvary Lajos, Apr 30 2009
(Magma) [ n eq 1 select 2 else n eq 2 select 3 else 3*Self(n-1)+Self(n-2): n in [1..30] ]; // Vincenzo Librandi, Aug 20 2011
(Haskell)
a006497 n = a006497_list !! n
a006497_list = 2 : 3 : zipWith (+) (map (* 3) $ tail a006497_list) a006497_list
-- Reinhard Zumkeller, Feb 19 2011
(PARI) my(x='x+O('x^30)); Vec((2-3*x)/(1-3*x-x^2)) \\ G. C. Greubel, Jul 05 2017
(PARI) apply( {A006497(n)=[2, 3]*([0, 1; 1, 3]^n)[, 1]}, [0..30]) \\ M. F. Hasler, Mar 06 2020

Crossrefs

Cf. A006190, A100230, A001622, A014176, A080039, A098316.

Sequence in context: A057838 A219497 A348126 * A038912 A019361 A093804

Adjacent sequences: A006494 A006495 A006496 * A006498 A006499 A006500

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

Definition completed by M. F. Hasler, Mar 06 2020

Status

approved