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A013655 a(n) = F(n+1) + L(n), where F(n) and L(n) are Fibonacci and Lucas numbers, respectively. 23
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

LINKS

Table of n, a(n) for n=0..39.

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).

Tanya Khovanova, Recursive Sequences

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

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

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.

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 August 20 22:35 EDT 2017. Contains 290837 sequences.