A047946 a(n) = 5*F(n)^2 + 3*(-1)^n where F(n) are the Fibonacci numbers A000045.

3, 2, 8, 17, 48, 122, 323, 842, 2208, 5777, 15128, 39602, 103683, 271442, 710648, 1860497, 4870848, 12752042, 33385283, 87403802, 228826128, 599074577, 1568397608, 4106118242, 10749957123, 28143753122, 73681302248, 192900153617, 505019158608, 1322157322202

Offset: 0

Author

John W. Layman, May 21 1999

Keywords

• nonn
• easy

Comments

Form the matrix A=[1,1,1;2,1,0;1,0,0]. a(n)=trace(A^n). - Paul Barry, Sep 22 2004

The set of prime divisors of elements of this sequence with the exception of 3 is the set of primes that do not divide odd Fibonacci numbers. - Tanya Khovanova, May 19 2008

If a(n) is prime then n is a power of 3 (Boase, 1998). The only values of k not exceeding 12 for which a(3^k) is prime are 0 and 1. - Amiram Eldar, Jun 19 2022

Links

• Mansur Boase, Problem 1558, Mathematics Magazine, Vol. 71, No. 4 (1998), p. 316; Primes in a Recursively Defined Sequence, Solution to Problem 1558 by TAMUK Problem Solvers, ibid., Vol. 72, No. 4 (1999), pp. 330-331.
• Tanya Khovanova, Divisibility of Odd Fibonaccis, 2008.
• Claudio de Jesús Pita Ruiz Velasco, On s-Fibonomials, Journal of Integer Sequences, Vol. 14 (2011), Article 11.3.7.
• Index entries for linear recurrences with constant coefficients, signature (2,2,-1).

Crossrefs

Cf. A000045, A000032, A005248.

Second row of array A028412.

Cf. A133247 (prime numbers p such that no odd Fibonacci number is divisible by p).

Programs

Mathematica

Table[LucasL[n]^2 - (-1)^n, {n, 0, 30}] (* Amiram Eldar, Feb 02 2022 *)

PARI

a(n)=5*fibonacci(n)^2+3*(-1)^n

Python

from sympy import fibonacci
def A047946(n): return 5*fibonacci(n)**2+(-3 if n&1 else 3) # Chai Wah Wu, Jul 29 2022

Formula

a(n) = F(3n)/F(n), n>0.

a(n) = 2*a(n-1)+2*a(n-2)-a(n-3).

a(n) = 3a(n-1)-a(n-2)+5(-1)^n.

a(n) = A005248(n) + (-1)^n.

G.f.: ( 3-4*x-2*x^2 ) / ( (1+x)*(x^2-3*x+1) ).

for n>0 a(n) = A000045(3n)/A000045(n) - Benoit Cloitre, Aug 30 2003

For n>0, the linear recurrence for the sequence F(n*k)^2 has signature (a(n),a(n),-1) for n odd, and (a(n),-a(n), 1) for n even. For example, the linear recurrence for the sequence F(3*k)^2 has signature (17, 17, -1) (conjectured). - Greg Dresden, Aug 30 2021

a(n) = Lucas(n)^2 - (-1)^n. - Amiram Eldar, Feb 02 2022

Extensions

Entry improved by comments from Michael Somos.