OFFSET
0,3
COMMENTS
Although the recurrence relation involves fractions, all the terms are integers.
The sequence of fractions b(n) = A350903(n)/A350904(n) is defined by the same recurrence relation, but with the initial terms 0 and 1 instead of 1 and 0.
André-Jeannin (1991) used this sequence and the sequence b(n) to prove that s = Sum_{n>=1} 1/F(n) (A079586) is an irrational number.
The sequence of ratios r(n) = b(n)/a(n) rapidly converges to s. For example, abs(r(16)-s) < 10^(-100) and abs(r(49)-s) < 10^(-1000).
LINKS
Amiram Eldar, Table of n, a(n) for n = 0..69
Richard André-Jeannin, Irrationalité de la somme des inverses de certaines suites récurrentes, Comptes Rendus de l'Académie des Sciences - Series I - Mathematics 308:19 (1989), pp. 539-541.
Richard André-Jeannin, Sequences of Integers Satisfying Recurrence Relations, The Fibonacci Quarterly, Vol. 29, No. 3 (1991), pp. 205-208.
MATHEMATICA
With[{F = Fibonacci, L = LucasL}, a[0] = 1; a[1] = 0; a[n_] := a[n] = (5*F[n]*F[n - 1]*F[2*n - 1]*a[n - 1] + F[n - 1]*L[n]*a[n - 2])/(L[n - 1]*F[n]); Array[a, 15, 0]]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Jan 21 2022
STATUS
approved