OFFSET
0,1
COMMENTS
a(n+1)/a(n) converges to 3 + sqrt 10.
For n > 0, the nearest integer to x^n for x=3+sqrt(10) - Lee A. Newberg, Oct 31 2025
a(0)/a(1) = 1/3 = [3]; a(1)/a(2) = 6/38 = [6,3]; a(2)/a(3) = 38/234 = [6,6,3], a(3)/a(4) = 234/1442 = [6,6,6,3]; a(4)/a(5) = 1442/8886 = [6,6,6,6,3];...etc. Lim a(n)/a(n+1) as n approaches infinity = 0.1622776...= 1/(3 + sqrt10) = sqrt(10) - 3.
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 Properties, The Journal of Mathematics and Computer Science (JMCS), Volume 8, Issue 1, Pages 81-92; sequence L_{6,n}
Tanya Khovanova, Recursive Sequences.
Vladimir V. Kruchinin and Maria Y. Perminova, Identities and Hadamard Product of the Generalized Fibonacci, Lucas, Catalan, and Harmonic Numbers, Journal of Integer Sequences, Vol. 28 (2025), Article 25.8.8. See p. 5.
Index entries for linear recurrences with constant coefficients, signature (6,1).
FORMULA
O.g.f.: 2*(-1+3*x)/(-1+6*x+x^2). - R. J. Mathar, Dec 02 2007
a(n) = 2*A005667(n). - R. J. Mathar, Nov 10 2013
E.g.f.: 2*exp(3*x)*cosh(sqrt(10)*x). - Stefano Spezia, Dec 21 2025
EXAMPLE
a(4) = 6*a(3)+a(2) = 6*234+38 = 1442.
MATHEMATICA
RecurrenceTable[{a[0] == 2, a[1] == 6, a[n] == 6 a[n-1] + a[n-2]}, a, {n, 30}] (* Vincenzo Librandi, Sep 19 2016 *)
LinearRecurrence[{6, 1}, {2, 6}, 30] (* G. C. Greubel, Nov 07 2018 *)
PROG
(Magma) I:=[2, 6]; [n le 2 select I[n] else 6*Self(n-1)+Self(n-2): n in [1..30]]; // Vincenzo Librandi, Sep 19 2016
(PARI) my(x='x+O('x^30)); Vec(2*(1-3*x)/(1-6*x-x^2)) \\ G. C. Greubel, Nov 07 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Gary W. Adamson, Jul 01 2003
EXTENSIONS
Edited and extended by Henry Bottomley, Jul 13 2003
STATUS
approved