A115339 a(2n-1)=F(n+1), a(2n)=L(n), where F(n) and L(n) are the Fibonacci and the Lucas sequences.

1, 1, 2, 3, 3, 4, 5, 7, 8, 11, 13, 18, 21, 29, 34, 47, 55, 76, 89, 123, 144, 199, 233, 322, 377, 521, 610, 843, 987, 1364, 1597, 2207, 2584, 3571, 4181, 5778, 6765, 9349, 10946, 15127, 17711, 24476, 28657, 39603, 46368, 64079, 75025, 103682, 121393, 167761

Offset: 1

Author

Giuseppe Coppoletta, Mar 06 2006

Keywords

• easy
• nonn

Comments

Alternate Fibonacci and Lucas sequence respecting their natural order.

See A116470 for an essentially identical sequence.

The ratio a(n+1)/a(n) increasingly approximates two constants: (5-sqrt(5))/2 (A094874) and (5+3*sqrt(5))/10 (A176015) according to whether n is odd or even. - Davide Rotondo, Oct 27 2024

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Fibonacci Number
• Eric Weisstein's World of Mathematics, Lucas Number.
• Index entries for linear recurrences with constant coefficients, signature (0,1,0,1).

Crossrefs

Cf. A000045, A000032.

Cf. A000930.

Cf. A094874, A176015.

Programs

Haskell

a115339 n = a115339_list !! (n-1)
a115339_list = [1, 1, 2, 3] ++
               zipWith (+) a115339_list (drop 2 a115339_list)
-- Reinhard Zumkeller, Aug 03 2013

Mathematica

f[n_] := If[OddQ@n, Fibonacci[(n + 3)/2], Fibonacci[n/2 - 1] + Fibonacci[n/2 + 1]]; Array[f, 50] (* Robert G. Wilson v, Apr 29 2006 *)

PARI

x='x+O('x^50); Vec(x*(-1-x-x^2-2*x^3)/(-1+x^2+x^4)) \\ G. C. Greubel, Apr 27 2017

Formula

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

G.f.: x*( -1-x-x^2-2*x^3 ) / ( -1+x^2+x^4 ). - R. J. Mathar, Mar 08 2011

Extensions

More terms from Robert G. Wilson v, Apr 29 2006