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

Author

Alexander Adamchuk, Aug 13 2006

Keywords

• nonn
• easy

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).

Crossrefs

Union of A000045 and A000032.

Cf. A288219 (even bisection).

Programs

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

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 *)

PARI

my(x='x+O('x^50)); 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

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