OFFSET
0,2
COMMENTS
a(n) = (t(i+4n) - t(i))/(t(i+2n+1) - t(i+2n-1)), where (t) is any sequence of the form t(n+2) = 8t(n+1) - 8t(n) + t(n-1) or of the form t(n+1) = 7t(n) - t(n-1) without regard to initial values as long as t(i+2n+1) - t(i+2n-1) != 0. - Klaus Purath, Jun 23 2024
LINKS
Indranil Ghosh, Table of n, a(n) for n = 0..597
Tanya Khovanova, Recursive Sequences
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)=0, a(1)=7.
G.f.: 7*x/(1-47*x+x^2).
a(n) = A004187(2n).
a(n) = 7*A049668(n). - R. J. Mathar, Oct 26 2015
E.g.f.: 2*exp(47*x/2)*sinh(21*sqrt(5)*x/2)/(3*sqrt(5)). - Stefano Spezia, Dec 14 2019
EXAMPLE
a(2) = F(8 * 2) / 3 = F(16) / 3 = 987 / 3 = 329. - Indranil Ghosh, Feb 05 2017
MAPLE
with(combinat); seq( fibonacci(8*n)/3, n=0..20); # G. C. Greubel, Dec 14 2019
MATHEMATICA
Fibonacci[8(Range[20]-1)]/3 (* G. C. Greubel, Dec 14 2019 *)
LinearRecurrence[{47, -1}, {0, 7}, 20] (* Harvey P. Dale, Dec 27 2019 *)
PROG
(PARI) a(n) = fibonacci(8*n)/3; \\ Michel Marcus, Feb 05 2017
(Magma) [Fibonacci(8*n)/3: n in [0..20]]; // G. C. Greubel, Dec 14 2019
(Sage) [fibonacci(8*n)/3 for n in (0..20)] # G. C. Greubel, Dec 14 2019
(GAP) List([0..20], n-> Fibonacci(8*n)/3 ); # G. C. Greubel, Dec 14 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Better description and more terms from Michael Somos
STATUS
approved