OFFSET
0,1
COMMENTS
This is the Lucas sequence V(4,-1). - Bruno Berselli, Jan 08 2013
REFERENCES
Ralph P. Grimaldi, Fibonacci and Catalan Numbers: An Introduction, (2012). See Exercise 26 at p. 99.
LINKS
Indranil Ghosh, Table of n, a(n) for n = 0..1591
Pooja Bhadouria, Deepika Jhala and Bijendra Singh, Binomial Transforms of the k-Lucas Sequences and its Properties, The Journal of Mathematics and Computer Science (JMCS), Volume 8, Issue 1 (2014), pp. 81-92; sequence L_{4,n}.
H. H. Ferns, Problem B-115, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 5, No. 2 (1967), p. 202; Identities for F_{kn} and L{kn}, Solution to Problem B-115 by Stanley Rabinowitz, ibid., Vol. 6, No. 1 (1968), pp. 92-93.
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.
Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.
Roberto Tauraso, Some Congruences for Central Binomial Sums Involving Fibonacci and Lucas Numbers, JIS 19 (2016), Article 16.5.4.
Wikipedia, Lucas sequence: Specific names.
Index entries for linear recurrences with constant coefficients, signature (4,1).
FORMULA
G.f.: (2-4*x)/(1-4*x-x^2).
a(n) = 4*a(n-1) +a(n-2) with n>1, a(0)=2, a(1)=4.
a(n) = (2+sqrt(5))^n + (2-sqrt(5))^n.
a(n) = 2*A001077(n).
a(n) = A000032(3*n).
a(n) = Sum_{k=0..n} C(n,k)*Lucas(n+k). - Paul D. Hanna, Oct 19 2010
a(n) = Fibonacci(6*n)/Fibonacci(3*n), n>0. - Gary Detlefs, Dec 26 2010
From Peter Bala, Mar 22 2015: (Start)
a(n) = ( Fibonacci(3*n + 2*k) - F(3*n - 2*k) )/Fibonacci(2*k) for nonzero integer k.
a(n) = ( Fibonacci(3*n + 2*k + 1) + F(3*n - 2*k - 1) )/Fibonacci(2*k + 1) for arbitrary integer k. (End)
a(n) = [x^n] ( (1 + 4*x + sqrt(1 + 8*x + 20*x^2))/2 )^n for n >= 1. - Peter Bala, Jun 23 2015
a(n) = L(n)*(L(n-1)*L(n+1) + 2*(-1)^n). - J. M. Bergot, Feb 05 2016
From Peter Bala Oct 14 2019: (Start)
Sum_{n >= 1} 1/( a(n) + (-1)^(n+1)*20/a(n) ) = 3/16.
Sum_{n >= 1} (-1)^(n+1)/( a(n) + (-1)^(n+1)*20/a(n) ) = 1/16. (End)
a(n) = (15*Fibonacci(n)^2*Lucas(n) + Lucas(n)^3)/4 (Ferns, 1967). - Amiram Eldar, Feb 06 2022
E.g.f.: 2*exp(2*x)*cosh(sqrt(5)*x). - Stefano Spezia, Jan 18 2025
a(n) = L(n)*(L(2*n) - (-1)^n) (see Grimaldi). - Stefano Spezia, Nov 22 2025
a(n) = Sum_{k=0..n} 2^k * binomial(n, k) * Lucas(k). - Amiram Eldar, Jan 18 2026
Sum_{n>=0} (-1)^n/(a(n)*a(n+1)) = 1/(4*sqrt(5)). - Amiram Eldar, Apr 06 2026
EXAMPLE
a(4) = L(3 * 4) = L(12) = 322. - Indranil Ghosh, Feb 05 2017
MATHEMATICA
Table[LucasL[3*n], {n, 0, 100}] (* G. C. Greubel, Nov 07 2018 *)
PROG
(PARI) polsym(x^2-4*x-1, 100)
(PARI) a(n)=sum(k=0, n, binomial(n, k)*(fibonacci(n+k-1)+fibonacci(n+k+1))) \\ Paul D. Hanna, Oct 19 2010
(SageMath) [lucas_number2(n, 4, -1) for n in range(0, 23)] # Zerinvary Lajos, May 14 2009
(Magma) [Lucas(3*n) : n in [0..100]]; // Vincenzo Librandi, Apr 14 2011
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Erich Friedman
STATUS
approved