OFFSET
0,2
LINKS
Indranil Ghosh, Table of n, a(n) for n = 0..596
Tanya Khovanova, Recursive Sequences
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume
Index entries for linear recurrences with constant coefficients, signature (47,-1).
FORMULA
a(n) = 47*a(n-1) - a(n-2), n>1. a(0)=1, a(1)=48.
G.f.: (1+x)/(1-47*x+x^2).
From Peter Bala, Mar 23 2015: (Start)
a(n) = ( Fibonacci(8*n + 8 - 2*k) + Fibonacci(8*n + 2*k) )/( Fibonacci(8 - 2*k) + Fibonacci(2*k) ), for k an arbitrary integer.
a(n) = ( Fibonacci(8*n + 8 - 2*k - 1) - Fibonacci(8*n + 2*k + 1) )/( Fibonacci(8 - 2*k - 1) - Fibonacci(2*k + 1) ), for k an arbitrary integer.
The aerated sequence (b(n))n>=1 = [1, 0, 48, 0, 2255, 0, 105937, 0, ...] is a fourth-order linear divisibility sequence; that is, if n | m then b(n) | b(m). It is the case P1 = 0, P2 = -45, Q = -1 of the 3-parameter family of divisibility sequences found by Williams and Guy. See A100047 for the connection with Chebyshev polynomials. (End)
EXAMPLE
a(2) = F(8 * 2 + 4) / 3 = F(20) / 3 = 6765 / 3 = 2255. - Indranil Ghosh, Feb 04 2017
MATHEMATICA
CoefficientList[Series[(1+x)/(1-47x+x^2), {x, 0, 20}], x] (* Harvey P. Dale, Feb 18 2011 *)
Table[Fibonacci[8*n+4]/3, {n, 0, 30}] (* G. C. Greubel, Dec 02 2017 *)
PROG
(PARI) for(n=0, 30, print1(fibonacci(8*n+4)/3, ", ")) \\ G. C. Greubel, Dec 02 2017
(Magma) [Fibonacci(8*n+4)/3: n in [0..30]]; // G. C. Greubel, Dec 02 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Better description and more terms from Michael Somos
2 more terms from Indranil Ghosh, Feb 04 2017
STATUS
approved