A302126 Interleaved Fibonacci and Lucas numbers.

0, 2, 1, 1, 1, 3, 2, 4, 3, 7, 5, 11, 8, 18, 13, 29, 21, 47, 34, 76, 55, 123, 89, 199, 144, 322, 233, 521, 377, 843, 610, 1364, 987, 2207, 1597, 3571, 2584, 5778, 4181, 9349, 6765, 15127, 10946, 24476, 17711, 39603, 28657, 64079, 46368, 103682, 75025, 167761

Offset: 0

Author

Patrick D McLean, Apr 01 2018

Keywords

• nonn
• easy

Examples

			a(10) = Fibonacci(5) = 5;
a(11) = Lucas(5) = 11.
		

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• T. Crilly, Interleaving Integer Sequences, The Mathematical Gazette, Vol. 91, No. 520 (Mar., 2007), pp. 27-33.
• Wikipedia, Interleave sequence
• Index entries for linear recurrences with constant coefficients, signature (0,1,0,1).

Crossrefs

Interleaves A000045 and A000032.

Programs

GAP

Flat(List([1..25],n->[Fibonacci(n),Lucas(1,-1,n)[2]])); # Muniru A Asiru, Apr 02 2018

Maple

a:= n-> (<<0|1>, <1|1>>^iquo(n, 2, 'r'). <<2*r, 1>>)[1, 1]:
seq(a(n), n=0..60);  # Alois P. Heinz, Apr 23 2018

Mathematica

Table[{Fibonacci[n], LucasL[n]}, {n, 0, 25}] // Flatten
LinearRecurrence[{0, 1, 0, 1}, {0, 2, 1, 1}, 52]
Flatten@ Array[{LucasL@#, Fibonacci@#} &, 26, 0] (* or *)
CoefficientList[Series[(x^3 - x^2 - 2x)/(x^4 + x^2 - 1), {x, 0, 51}], x] (* Robert G. Wilson v, Apr 02 2018 *)

PARI

concat(0, Vec(x*(2 - x)*(1 + x) / (1 - x^2 - x^4) + O(x^60))) \\ Colin Barker, Apr 02 2018

Formula

a(0) = 0; a(1) = 2; a(2) = 1; a(3) = 1; a(n) = a(n-2) + a(n-4), n >= 4.

G.f.: x*(2 - x)*(1 + x) / (1 - x^2 - x^4). - Colin Barker, Apr 02 2018

a(0) = 0; a(1) = 2; a(2n) = (a(2n-1) + a(2n-2))/2; a(2n+1) = a(2n) + 2*a(2n-2), n >= 1. - Daniel Forgues, Jul 29 2018

Extensions

More terms from Colin Barker, Apr 02 2018