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

0,2

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).

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

Example

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

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 *)

Prog

(PARI) concat(0, Vec(x*(2 - x)*(1 + x) / (1 - x^2 - x^4) + O(x^60))) \\ Colin Barker, Apr 02 2018
(GAP) Flat(List([1..25], n->[Fibonacci(n), Lucas(1, -1, n)[2]])); # Muniru A Asiru, Apr 02 2018

Crossrefs

Interleaves A000045 and A000032.

Sequence in context: A363095 A057774 A089355 * A136043 A336420 A254055

Adjacent sequences: A302123 A302124 A302125 * A302127 A302128 A302129

Keyword

nonn,easy

Author

Patrick D McLean, Apr 01 2018

Extensions

More terms from Colin Barker, Apr 02 2018

Status

approved