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%I #139 Feb 14 2024 08:43:01
%S 3,2,5,7,12,19,31,50,81,131,212,343,555,898,1453,2351,3804,6155,9959,
%T 16114,26073,42187,68260,110447,178707,289154,467861,757015,1224876,
%U 1981891,3206767,5188658,8395425,13584083,21979508,35563591,57543099,93106690,150649789,243756479
%N a(n) = F(n+1) + L(n), where F(n) and L(n) are Fibonacci and Lucas numbers, respectively.
%C Apart from initial term, same as A001060.
%C Pisano period lengths same as for A001060. - _R. J. Mathar_, Aug 10 2012
%C The beginning of this sequence is the only sequence of four consecutive primes in a Fibonacci-type sequence. - _Franklin T. Adams-Watters_, Mar 21 2015
%C (a(2*k), a(2*k+1)) give for k >= 0 the proper positive solutions of one of two families (or classes) of solutions (x, y) of the indefinite binary quadratic form x^2 + x*y - y^2 of discriminant 5 representing 11. The other family of such solutions is given by (x2, y2) = (b(2*k), b(2*k+1)) with b = A104449. See the formula in terms of Chebyshev S polynomials S(n, 3) = A001906(n+1) below, which follows from the fundamental solution (3, 2) by applying positive powers of the automorphic matrix A^k = Matrix([A(k), B(k)], [B(k), A(k+1)]), with A(k) = S(k-1, 3) - S(k-2, 3) and B(k) = S(k-1, 3). See also A089270 with the Alfred Brousseau link with D = 11. - _Wolfdieter Lang_, May 28 2019
%H G. C. Greubel, <a href="/A013655/b013655.txt">Table of n, a(n) for n = 0..2500</a>
%H Mark W. Coffey, James L. Hindmarsh, Matthew C. Lettington, John Pryce, <a href="http://arxiv.org/abs/1502.03085">On Higher Dimensional Interlacing Fibonacci Sequences, Continued Fractions and Chebyshev Polynomials</a>, arXiv:1502.03085 [math.NT], 2015 (see p. 31).
%H Rigoberto Flórez, Robinson A. Higuita, Antara Mukherjee, <a href="https://arxiv.org/abs/1804.02481">The Geometry of some Fibonacci Identities in the Hosoya Triangle</a>, arXiv:1804.02481 [math.NT], 2018.
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H Shaoxiong Yuan, <a href="https://arxiv.org/abs/1907.12459">Generalized Identities of Certain Continued Fractions</a>, arXiv:1907.12459 [math.NT], 2019.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,1).
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F a(n) = A000045(n+1) + A000032(n).
%F a(n) = a(n-1) + a(n-2).
%F a(n) = F(n+3) - F(n-2) for n>1, where F=A000045. - _Gerald McGarvey_, Jul 10 2004
%F a(n) = 2*F(n-3) + F(n) for n>1. - _Zerinvary Lajos_, Oct 05 2007
%F G.f.: (3-x)/(1-x-x^2). - _Philippe Deléham_, Nov 19 2008
%F a(n) = Sum_{k = n-3..n+1} F(k). - _Gary Detlefs_, Dec 30 2012
%F a(n) = ((3*sqrt(5)+1)*(((1+sqrt(5))/2)^n)+(3*sqrt(5)-1)*(((1-sqrt(5))/2)^n))/(2*sqrt(5)). - _Bogart B. Strauss_, Jul 19 2013
%F a(n) = F(n+3) + F(n-3) - 3*F(n) for n>1. - _Bruno Berselli_, Dec 29 2016
%F Bisection: a(2*k) = 3*S(k, 3) - 4*S(k-1, 3), a(2*k+1) = 2*S(k, 3) + S(k-1, 3), for k >= 0, with the Chebyshev S(n, 3) polynomials from A001906(n+1) for n >= -1. - _Wolfdieter Lang_, May 28 2019
%F a(3n + 2)/a(3n - 1) = continued fraction 4,4,4,...,4,-5 (that's n 4's followed by a single -5). - _Greg Dresden_ and _Shaoxiong Yuan_, Jul 16 2019
%F E.g.f.: ((- 1 + 3*sqrt(5))*exp((1/2)*(1 - sqrt(5))*x) + (1 + 3*sqrt(5))*exp((1/2)*(1 + sqrt(5))*x))/(2*sqrt(5)). - _Stefano Spezia_, Jul 17 2019
%F a(n) = (F(3n+1) - F(n+1)^3)/(F(n)^2) for n>1, where F(n) = A000045(n). - _Michael Tulskikh_, Jul 22 2020
%F a(n) = 3 * Sum_{k=0..n-2} A168561(n-2,k) + 2 * Sum_{k=0..n-1} A168561(n-1,k), n>0. - _R. J. Mathar_, Feb 14 2024
%p with(combinat): a:=n->2*fibonacci(n-1)+fibonacci(n+2): seq(a(n), n=0..40); # _Zerinvary Lajos_, Oct 05 2007
%t LinearRecurrence[{1, 1}, {3, 2}, 40] (* or *)
%t Table[Fibonacci[n + 1] + LucasL[n], {n, 0, 40}] (* or *)
%t Table[Fibonacci[n + 3] + Fibonacci[n - 3] - 3*Fibonacci[n], {n,2,40}] (* _Bruno Berselli_, Dec 30 2016 *)
%o (Magma) [2*Fibonacci(n-3)+Fibonacci(n): n in [2..41]]; // _Vincenzo Librandi_, Apr 16 2011
%o (Magma) [GeneralizedFibonacciNumber(3, 2, n): n in [0..39]]; // _Arkadiusz Wesolowski_, Mar 16 2016
%o (PARI) a(n)=([0,1; 1,1]^n*[3;2])[1,1] \\ _Charles R Greathouse IV_, Sep 24 2015
%o (PARI) a(n)=2*fibonacci(n-3) + fibonacci(n) \\ _Charles R Greathouse IV_, Sep 24 2015
%Y Cf. A000045, A001060, A001906, A089270, A104449.
%K nonn,easy
%O 0,1
%A _Mohammad K. Azarian_
%E Definition corrected by _Gary Detlefs_, Dec 30 2012
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