A088545 - OEIS

A088545

Quotient Fibonacci(5*n)/(5*Fibonacci(n)), where Fibonacci(n) = A000045(n).

1, 11, 61, 451, 3001, 20801, 141961, 974611, 6675901, 45768251, 313671601, 2150012161, 14736206161, 101003973851, 692290189501, 4745031073651, 32522917584361, 222915417520961, 1527884938291801, 10472279325329251, 71778069881360701, 491974211042344811, 3372041404278257761

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OFFSET

1,2

COMMENTS

The sequences {Fibonacci(k*n)/(Fibonacci(k)*Fibonacci(n)): n >= 1} are integral in the three cases k = 1 (A000012), k = 2 (A000032) and k = 5 (the present sequence). See Young, Section 4. - Peter Bala, Jan 09 2023

LINKS

Table of n, a(n) for n=1..23.

Chatchawan Panraksa and Aram Tangboonduangjit, On Some Arithmetic Properties of a Sequence Related to the Quotient of Fibonacci Numbers, Fibonacci Quart., Vol. 55, No. 1 (2017), 21-28.

Paul Thomas Young, p-adic congruences for generalized Fibonacci sequences, The Fibonacci Quarterly, Vol. 32, No. 1 (1994), 2-10.

FORMULA

a(n) = 5*Fib(n)^2*(Fib(n)^2 + (-1)^n) + 1 = 5*A007598(n)*A059929(n+1) + 1.

a(n) = A103326(n) / 5.

G.f.: -x*(x^4-4*x^3-9*x^2+6*x+1) / ((x-1)*(x^2-7*x+1)*(x^2+3*x+1)). - Colin Barker, Jul 16 2013

The expansion of exp(Sum_{n >= 1} a(n)*x^n/n) = 1 + x + 6*x^2 + 26*x^3 + 151*x^4 + 851*x^5 + 5101*x^6 + ... has integral coefficients and is equal to G(x)^(1/5), where G(x) is the o.g.f. of A001656. See Young, Theorem 3. - Peter Bala, Jan 09 2023

From Amiram Eldar, Jan 10 2026: (Start)

Formulas from Panraksa and Tangboonduangjit (2017):

gcd(a(n), Fibonacci(n)) = 1.

a(n) == 1 (mod 10).

p == 1 (mod 10) for prime p that divide a(n) for some n.

Prime p | a(n) for some n if and only if p != 5 and 5 | A001177(p).

Prime p | a(n) for some n if and only if p != 5, A001177(p) | 5*n and A112765(A001177(p)) = A112765(n) + 1.

gcd(a(m), a(n)) = a(gcd(m, n)) if A112765(m) = A112765(n), and 1 otherwise. (End)

MATHEMATICA

a[n_] := Fibonacci[5*n]/(5*Fibonacci[n]); Array[a, 25] (* Amiram Eldar, Jan 10 2026 *)

PROG

(PARI) a(n)=fibonacci(5*n)/(5*fibonacci(n)); \\ Joerg Arndt, Jul 16 2013