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A006497 a(n) = 3a(n-1) + a(n-2).
(Formerly M0910)
16
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 (list; graph; refs; listen; history; text; internal format)
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

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; http://www.hyle.org.

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

Tanya Khovanova, Recursive Sequences

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

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

Index entries for sequences related to Chebyshev polynomials.

Index to sequences with linear recurrences with constant coefficients, signature (3,1).

FORMULA

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

a(n) = [(3 + sqrt13)/2]^n + [(3 - sqrt13)/2]^n; A006190(n-2) + A006190(n) = a(n-1); [a(n)]^2 - 13[A006190(n)]^2 = 4(-1)^n. - Gary W. Adamson, Jun 15 2003

E.g.f. : 2exp(3x/2)cosh(sqrt(13)x/2); a(n)=2^(1-n)sum{k=0..floor(n/2), C(n, 2k)13^k3^(n-2k)}. a(n)=2T(n, 3i/2)(-i)^n with T(n, x) Chebyshev's polynomials of the first kind (see A053120) and i^2=-1. - Paul Barry, Nov 15 2003

Comments 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)

Contribution 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(a(n+k)/a(k), k=infinity) = (a(n) + A006190(n)*sqrt(13))/2.

Limit(a(n)/A006190(n), n=infinity) = 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

MATHEMATICA

Table[LucasL[n, 3], {n, 0, 26}] [Zerinvary Lajos, Jul 09 2009]

PROG

(Sage) [lucas_number2(n, 3, -1) for n in xrange(0, 25)]# [From 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..40] ]; // 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

CROSSREFS

Cf. A006190.

Cf. A100230.

Cf. A001622, A014176, A080039, A098316.

Sequence in context: A159458 A057838 A219497 * A038912 A019361 A093804

Adjacent sequences:  A006494 A006495 A006496 * A006498 A006499 A006500

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified November 27 18:17 EST 2014. Contains 250249 sequences.