OFFSET
0,4
COMMENTS
Let f(x, y) = ((y+1)^x - (y-1)^x * cos(Pi*x))/(y * 2^x). Then f(n, sqrt(3)) are the rational numbers a(n)/A060723(n) and f(n, sqrt(5)) the Fibonacci numbers A000045(n).
From Paul Curtz, Dec 05 2018: (Start)
The binomial inverse of the rational sequence r(n) starts 0, 1, -1, 3/2, -2, 11/4, -15/4, 41/8, -7, 153/16, -209/16, ... and is up to signs equal to r(n). The difference table starts:
0, 1, 1, 3/2, 2, 11/4, 15/4, 41/8, ...
1, 0, 1/2, 1/2, 3/4, 1, 11/8, 15/8, ...
-1, 1/2, 0, 1/4, 1/4, 3/8, 1/2, 11/16, ...
3/2, -1/2, 1/4, 0, 1/8, 1/8, 3/16, 1/4, ...
...
FORMULA
A recurrence for r(n) is given in A060723.
MATHEMATICA
Table[Numerator[Simplify[((1/2 (Sqrt[3] + 1))^x - (1/2 (Sqrt[3] - 1))^x Cos[Pi x])/Sqrt[3]]], {x, 0, 36}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Peter Luschny, Jun 02 2018
STATUS
approved