A396115 a(n) = F((n-1)*(n+1))/(F(n-1)*F(n+1)), where F(n) is the n-th Fibonacci number A000045(n).

1, 2, 1, 7, 61, 1932, 141961, 28616232, 14835905701, 20274223678327, 72347321822391001, 676560511993808919302, 16557761623387534111504801, 1061047233926922264334773774782, 177999626303093264515509668491829161, 78178603961589153039675223631587252222807

Offset

0,2

Links

Alois P. Heinz, Table of n, a(n) for n = 0..70

Wikipedia, Cassini and Catalan identities.

Formula

a(n) = F((n-1)*(n+1))/(F(n-1)*F(n+1)) with F(n) = (phi^n - (1-phi)^n)/sqrt(5) and phi = (1 + sqrt(5))/2 = A001622. a(1) = 2 is found by taking the limit of the quotient for n->1.

a(n) = a(-n), this follows from the extension above.

a(n) mod 2 = A093719(n+3).

a(n) mod 5 = A000034(n).

From Peter Luschny, May 24 2026: (Start)

a(n) = F(n^2 - 1) / (F(n)^2 + (-1)^n).

a(n) = (phi^(n^2 - 1) - psi^(n^2 - 1))*sqrt(5)/(((phi^n - psi^n)^2 + 5*(-1)^n)) where phi = (1 + sqrt(5))/2 and psi = 1 - phi, for n != 1. (End)

Maple

a:= n-> `if`(abs(n)=1, 2, (f->f((n-1)*(n+1))/(f(n-1)*f(n+1)))(k->(<<0|1>, <1|1>>^k)[1, 2])):
seq(a(n), n=0..15);
# Alternative: uses only two Fibonacci evaluations per call.
f := combinat:-fibonacci: a := n -> local h; if abs(n) = 1 then 2 else
h := f(n); f(n * n - 1) / (h * h + ifelse(n::odd, -1, 1)) fi:
seq(a(n), n = 0..15); # Peter Luschny, May 24 2026

Mathematica

A396115[n_] := If[n == 1, 2, Fibonacci[n^2 - 1]/(Fibonacci[n]^2 + (-1)^n)];
Array[A396115, 20, 0] (* Paolo Xausa, May 25 2026 *)

Prog

(Python)
def fibp(n: int) -> tuple[int, int]:
    if n == 0: return (0, 1)
    a, b = fibp(n >> 1)
    c = a * (2 * b - a)
    d = a * a + b * b
    return (d, c + d) if (n & 1) else (c, d)
def a(n: int) -> int:
    if n < 2: return n + 1
    f = fibp(n)[0]
    ff = fibp(n * n - 1)[0]
    return ff // (f * f + (-1 if n & 1 else 1))
A = [a(n) for n in range(17)]; print(A)  # Peter Luschny, May 24 2026

Crossrefs

Cf. A000034, A000045, A001622, A002163, A051294, A059929, A093719, A103624, A205505, A396233, A396358.

Sequence in context: A183272 A138346 A085073 * A131288 A111789 A179447

Adjacent sequences: A396109 A396111 A396114 * A396116 A396117 A396118

Keyword

nonn,easy

Author

Alois P. Heinz, May 17 2026

Status

approved