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

%I A047946 #45 Dec 30 2023 23:47:31
%S A047946 3,2,8,17,48,122,323,842,2208,5777,15128,39602,103683,271442,710648,
%T A047946 1860497,4870848,12752042,33385283,87403802,228826128,599074577,
%U A047946 1568397608,4106118242,10749957123,28143753122,73681302248,192900153617,505019158608,1322157322202
%N A047946 a(n) = 5*F(n)^2 + 3*(-1)^n where F(n) are the Fibonacci numbers A000045.
%C A047946 Form the matrix A=[1,1,1;2,1,0;1,0,0]. a(n)=trace(A^n). - _Paul Barry_, Sep 22 2004
%C A047946 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
%C A047946 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
%H A047946 Mansur Boase, <a href="http://www.jstor.org/stable/2690711">Problem 1558</a>, Mathematics Magazine, Vol. 71, No. 4 (1998), p. 316; <a href="https://www.jstor.org/stable/2691234">Primes in a Recursively Defined Sequence</a>, Solution to Problem 1558 by TAMUK Problem Solvers, ibid., Vol. 72, No. 4 (1999), pp. 330-331.
%H A047946 Tanya Khovanova, <a href="http://blog.tanyakhovanova.com/?p=25">Divisibility of Odd Fibonaccis</a>, 2008.
%H A047946 Claudio de Jesús Pita Ruiz Velasco, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Pita/pita12.html">On s-Fibonomials</a>, Journal of Integer Sequences, Vol. 14 (2011), Article 11.3.7.
%H A047946 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-1).
%F A047946 a(n) = F(3n)/F(n), n>0.
%F A047946 a(n) = 2*a(n-1)+2*a(n-2)-a(n-3).
%F A047946 a(n) = 3a(n-1)-a(n-2)+5(-1)^n.
%F A047946 a(n) = A005248(n) + (-1)^n.
%F A047946 G.f.: ( 3-4*x-2*x^2 ) / ( (1+x)*(x^2-3*x+1) ).
%F A047946 for n>0 a(n) = A000045(3n)/A000045(n) - _Benoit Cloitre_, Aug 30 2003
%F A047946 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
%F A047946 a(n) = Lucas(n)^2 - (-1)^n. - _Amiram Eldar_, Feb 02 2022
%t A047946 Table[LucasL[n]^2 - (-1)^n, {n, 0, 30}] (* _Amiram Eldar_, Feb 02 2022 *)
%o A047946 (PARI) a(n)=5*fibonacci(n)^2+3*(-1)^n
%o A047946 (Python)
%o A047946 from sympy import fibonacci
%o A047946 def A047946(n): return 5*fibonacci(n)**2+(-3 if n&1 else 3) # _Chai Wah Wu_, Jul 29 2022
%Y A047946 Cf. A000045, A000032, A005248.
%Y A047946 Second row of array A028412.
%Y A047946 Cf. A133247 (prime numbers p such that no odd Fibonacci number is divisible by p).
%K A047946 nonn,easy
%O A047946 0,1
%A A047946 _John W. Layman_, May 21 1999
%E A047946 Entry improved by comments from _Michael Somos_.