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A047946
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a(n) = 5*F(n)^2 + 3*(-1)^n where F(n) are the Fibonacci numbers A000045.
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5
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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
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OFFSET
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0,1
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COMMENTS
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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
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LINKS
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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.
Claudio de Jesús Pita Ruiz Velasco, On s-Fibonomials, Journal of Integer Sequences, Vol. 14 (2011), Article 11.3.7.
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FORMULA
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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.
G.f.: ( 3-4*x-2*x^2 ) / ( (1+x)*(x^2-3*x+1) ).
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
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MATHEMATICA
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Table[LucasL[n]^2 - (-1)^n, {n, 0, 30}] (* Amiram Eldar, Feb 02 2022 *)
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PROG
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(PARI) a(n)=5*fibonacci(n)^2+3*(-1)^n
(Python)
from sympy import fibonacci
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CROSSREFS
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Cf. A133247 (prime numbers p such that no odd Fibonacci number is divisible by p).
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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