A116470 All distinct Fibonacci and Lucas numbers.

0, 1, 2, 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

0,3

Comments

See A115339 for an essentially identical sequence.

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Index entries for linear recurrences with constant coefficients, signature (0,1,0,1).

Formula

a(0) = 0, a(1) = 1, a(2) = 2, a(3) = 3, a(4) = 4, a(5) = 5, a(6) = 7, a(n) = a(n-2) + a(n-4) for n>6.

a(2*n) = Lucas(n+1) = Fibonacci(n) + Fibonacci(n+2) for n>1.

a(2*n+1) = Fibonacci(n+3) for n>2.

G.f.: -x*(x^2+x+1)*(x^3+x+1)/(-1+x^4+x^2). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 12 2009

a(n) = FL((n + 2 + 3*(n mod 2))/2, n mod 2, 1/2)) for n >= 3. Here FL(n, a, b) = hypergeom([a - n/2, b - n/2], [1 - n], -4). - Peter Luschny, Sep 03 2019

Maple

FL := (n, a, b) -> hypergeom([a - n/2, b - n/2], [1 - n], -4):
a := n -> `if`(n < 3, n, FL((n + 2 + 3*irem(n, 2))/2, irem(n, 2), 1/2)):
seq(simplify(a(n)), n=0..52); # Peter Luschny, Sep 03 2019

Mathematica

CoefficientList[Series[-x*(x^2 + x + 1)*(x^3 + x + 1)/(-1 + x^4 + x^2), {x, 0, 50}], x] (* G. C. Greubel, Dec 21 2017 *)
With[{nn=50}, Select[Union[Join[LucasL[Range[0, nn]], Fibonacci[Range[0, nn]]]], #<=200000&]] (* Harvey P. Dale, Jul 05 2019 *)

Prog

(Haskell)
import Data.List (transpose)
a116470 n = a116470_list !! n
a116470_list = 0 : 1 : 2 : concat
               (transpose [drop 4 a000045_list, drop 3 a000032_list])
-- Reinhard Zumkeller, Aug 03 2013
(PARI) x='x+O('x^30); concat([0], Vec(-x*(x^2+x+1)*(x^3+x+1)/( -1+x^4 +x^2))) \\ G. C. Greubel, Dec 21 2017
(PARI) a(n)=if(n<6, n, if(n%2, fibonacci(n\2+3), fibonacci(n\2)+fibonacci(n\2+2))) \\ Charles R Greathouse IV, Oct 14 2021

Crossrefs

Union of A000045 and A000032.

Cf. A288219 (even bisection).

Sequence in context: A199120 A118083 A241093 * A115649 A191168 A309879

Adjacent sequences: A116467 A116468 A116469 * A116471 A116472 A116473

Keyword

nonn,easy

Author

Alexander Adamchuk, Aug 13 2006

Status

approved