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A013655 a(n) = F(n+1) + L(n), where F(n) and L(n) are Fibonacci and Lucas numbers, respectively. 24
3, 2, 5, 7, 12, 19, 31, 50, 81, 131, 212, 343, 555, 898, 1453, 2351, 3804, 6155, 9959, 16114, 26073, 42187, 68260, 110447, 178707, 289154, 467861, 757015, 1224876, 1981891, 3206767, 5188658, 8395425, 13584083, 21979508, 35563591, 57543099, 93106690, 150649789, 243756479 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Apart from initial term, same as A001060.

Pisano period lengths same as for A001060. - R. J. Mathar, Aug 10 2012

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

(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

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..2500

Mark W. Coffey, James L. Hindmarsh, Matthew C. Lettington, John Pryce, On Higher Dimensional Interlacing Fibonacci Sequences, Continued Fractions and Chebyshev Polynomials, arXiv:1502.03085 [math.NT], 2015 (see p. 31).

Rigoberto Flórez, Robinson A. Higuita, Antara Mukherjee, The Geometry of some Fibonacci Identities in the Hosoya Triangle, arXiv:1804.02481 [math.NT], 2018.

Tanya Khovanova, Recursive Sequences

Shaoxiong Yuan, Generalized Identities of Certain Continued Fractions, arXiv:1907.12459 [math.NT], 2019.

Index entries for linear recurrences with constant coefficients, signature (1,1)

Index entries for sequences related to Chebyshev polynomials.

FORMULA

a(n) = A000045(n+1) + A000032(n).

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

a(n) = F(n+3) - F(n-2) for n>1, where F=A000045. - Gerald McGarvey, Jul 10 2004

a(n) = 2*F(n-3) + F(n) for n>1. - Zerinvary Lajos, Oct 05 2007

G.f.: (3-x)/(1-x-x^2). - Philippe Deléham, Nov 19 2008

a(n) = Sum_{k = n-3..n+1} F(k). - Gary Detlefs, Dec 30 2012

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

a(n) = F(n+3) + F(n-3) - 3*F(n) for n>1. - Bruno Berselli, Dec 29 2016

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

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

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

MAPLE

with(combinat): a:=n->2*fibonacci(n-1)+fibonacci(n+2): seq(a(n), n=0..40); # Zerinvary Lajos, Oct 05 2007

MATHEMATICA

LinearRecurrence[{1, 1}, {3, 2}, 40] (* or *)

Table[Fibonacci[n + 1] + LucasL[n], {n, 0, 40}] (* or *)

Table[Fibonacci[n + 3] + Fibonacci[n - 3] - 3*Fibonacci[n], {n, 2, 40}] (* Bruno Berselli, Dec 30 2016 *)

PROG

(MAGMA) [2*Fibonacci(n-3)+Fibonacci(n): n in [2..41]]; // Vincenzo Librandi, Apr 16 2011

(MAGMA) [GeneralizedFibonacciNumber(3, 2, n): n in [0..39]]; // Arkadiusz Wesolowski, Mar 16 2016

(PARI) a(n)=([0, 1; 1, 1]^n*[3; 2])[1, 1] \\ Charles R Greathouse IV, Sep 24 2015

(PARI) a(n)=2*fibonacci(n-3) + fibonacci(n) \\ Charles R Greathouse IV, Sep 24 2015

CROSSREFS

Cf. A000045, A001060, A001906, A089270, A104449.

Sequence in context: A110338 A171018 A239260 * A223701 A220519 A094894

Adjacent sequences:  A013652 A013653 A013654 * A013656 A013657 A013658

KEYWORD

nonn,easy

AUTHOR

Mohammad K. Azarian

EXTENSIONS

Definition corrected by Gary Detlefs, Dec 30 2012

STATUS

approved

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Last modified October 16 07:30 EDT 2019. Contains 328051 sequences. (Running on oeis4.)