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A047946 a(n) = 5*F(n)^2 + 3*(-1)^n where F(n) are the Fibonacci numbers A000045. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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

Table of n, a(n) for n=0..29.

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).

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

a(n) = (3/2+(1/2)*sqrt(5))^n + (-1)^n + (3/2-(1/2)*sqrt(5))^n, with n>=0 - Paolo P. Lava, Jun 12 2008

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

MATHEMATICA

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

PROG

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

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).

Sequence in context: A171634 A107300 A285787 * A066045 A110866 A257958

Adjacent sequences:  A047943 A047944 A047945 * A047947 A047948 A047949

KEYWORD

nonn,easy

AUTHOR

John W. Layman, May 21 1999

EXTENSIONS

Entry improved by comments from Michael Somos.

STATUS

approved

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Last modified July 6 12:26 EDT 2022. Contains 355110 sequences. (Running on oeis4.)