OFFSET
0,2
COMMENTS
Let G and H be sequences of the form G(i) = 4*G(i-1) - G(i-2) and H(j) = 14*H(j-1) - H(j-2) for arbitrary integers i, j and without regard to initial values of G or H, then a(n) = (G(i) + G(i+8*n+4))/(14*G(i+4*n+2)) = (H(j) + H(j+4*n+2))/(14*H(j+2*n+1)) with the exception of G(i+4*n+2) or H(j+2*n+1) != 0. - Klaus Purath, Aug 31 2020
LINKS
Colin Barker, Table of n, a(n) for n = 0..400
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (194,-1).
FORMULA
G.f.: (1 - x)/(1 - 194*x + x^2).
a(n) = a(-1-n).
a(n) = cosh((2*n + 1)*arccosh(7))/7.
a(n) = ((7 + 4*sqrt(3))^(2*n + 1) + 1/(7 + 4*sqrt(3))^(2*n + 1))/14.
a(n) = (a(n-1)^2 + 192)/a(n-2). - Klaus Purath, Aug 31 2020
a(n) = (1/7)*T(2*n+1, 7), where T(n,x) denotes the n-th Chebyshev polynomial of the first kind. - Peter Bala, Jul 08 2022
MATHEMATICA
LinearRecurrence[{194, -1}, {1, 193}, 20]
PROG
(PARI) x='x+O('x^99); Vec((1-x)/(1-194*x+x^2)) \\ Altug Alkan, Apr 06 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Bruno Berselli, Apr 05 2018
STATUS
approved