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

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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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).

Crossrefs

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

Programs

Haskell

a006497 n = a006497_list !! n
a006497_list = 2 : 3 : zipWith (+) (map (* 3) $ tail a006497_list) a006497_list
-- Reinhard Zumkeller, Feb 19 2011

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

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 *)

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

Sage

[lucas_number2(n,3,-1) for n in range(0, 30)] # Zerinvary Lajos, Apr 30 2009

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

a(2*n+1) = 3 + 3*Sum_{k=1..n} a(2*k). - Greg Dresden and Canran Wang, Jul 11 2024

From Peter Bala, Jul 14 2025: (Start)

The following series telescope (Cf. A000032):

For k >= 1, Sum_{n >= 1} (-1)^((k+1)*(n+1)) * a(2*n*k)/(a((2*n-1)*k)*a((2*n+1)*k)) = 1/a(k)^2.

For positive even k, Sum_{n >= 1} 1/(a(k*n) - (a(k) + 2)/a(k*n)) = 1/(a(k) - 2) and

Sum_{n >= 1} (-1)^(n+1)/(a(k*n) + (a(k) - 2)/a(k*n)) = 1/(a(k) + 2).

For positive odd k, Sum_{n >= 1} 1/(a(k*n) - (-1)^n*(a(2*k) + 2)/a(k*n)) = (a(k) + 2)/(2*(a(2*k) - 2)) and

Sum_{n >= 1} (-1)^(n+1)/(a(k*n) - (-1)^n*(a(2*k) + 2)/a(k*n)) = (a(k) - 2)/(2*(a(2*k) - 2)). (End)

Extensions

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