|
| |
|
|
A006497
|
|
a(n) = 3*a(n-1) + a(n-2).
(Formerly M0910)
|
|
20
|
|
|
|
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
|
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}.
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
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 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 + 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. (End)
From Paul Barry, Nov 15 2003: (Start)
E.g.f. : 2exp(3x/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(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
a(n) = [x^n] ( (1 + 3*x + sqrt(1 + 6*x + 13*x^2))/2 )^n for n >= 1. - Peter Bala, Jun 23 2015
|
|
|
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, 26}] (* Zerinvary Lajos, Jul 09 2009 *)
LucasL[Range[0, 20], 3] (* Eric W. Weisstein, Apr 17 2018 *)
|
|
|
PROG
|
(Sage) [lucas_number2(n, 3, -1) for n in xrange(0, 25)] # 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
(PARI) x='x+O('x^50); Vec((2-3*x)/(1-3*x-x^2)) \\ G. C. Greubel, Jul 05 2017
|
|
|
CROSSREFS
|
Cf. A006190, A100230, A001622, A014176, A080039, A098316.
Sequence in context: A305846 A057838 A219497 * A038912 A019361 A093804
Adjacent sequences: A006494 A006495 A006496 * A006498 A006499 A006500
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
N. J. A. Sloane
|
|
|
STATUS
|
approved
|
| |
|
|
