A302126 - OEIS

%I #40 Aug 08 2018 09:59:05

%S 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,

%T 322,233,521,377,843,610,1364,987,2207,1597,3571,2584,5778,4181,9349,

%U 6765,15127,10946,24476,17711,39603,28657,64079,46368,103682,75025,167761

%N Interleaved Fibonacci and Lucas numbers.

%H Colin Barker, <a href="/A302126/b302126.txt">Table of n, a(n) for n = 0..1000</a>

%H T. Crilly, <a href="http://www.jstor.org/stable/40378281">Interleaving Integer Sequences</a>, The Mathematical Gazette, Vol. 91, No. 520 (Mar., 2007), pp. 27-33.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Interleave_sequence">Interleave sequence</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,0,1).

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

%F G.f.: x*(2 - x)*(1 + x) / (1 - x^2 - x^4). - _Colin Barker_, Apr 02 2018

%F 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

%e a(10) = Fibonacci(5) = 5;

%e a(11) = Lucas(5) = 11.

%p a:= n-> (<<0|1>, <1|1>>^iquo(n, 2, 'r'). <<2*r, 1>>)[1, 1]:

%p seq(a(n), n=0..60);  # _Alois P. Heinz_, Apr 23 2018

%t Table[{Fibonacci[n], LucasL[n]}, {n, 0, 25}] // Flatten

%t LinearRecurrence[{0, 1, 0, 1}, {0, 2, 1, 1}, 52]

%t Flatten@ Array[{LucasL@#, Fibonacci@#} &, 26, 0] (* or *)

%t CoefficientList[Series[(x^3 - x^2 - 2x)/(x^4 + x^2 - 1), {x, 0, 51}], x] (* _Robert G. Wilson v_, Apr 02 2018 *)

%o (PARI) concat(0, Vec(x*(2 - x)*(1 + x) / (1 - x^2 - x^4) + O(x^60))) \\ _Colin Barker_, Apr 02 2018

%o (GAP) Flat(List([1..25],n->[Fibonacci(n),Lucas(1,-1,n)[2]])); # _Muniru A Asiru_, Apr 02 2018

%Y Interleaves A000045 and A000032.

%K nonn,easy

%O 0,2

%A _Patrick D McLean_, Apr 01 2018

%E More terms from _Colin Barker_, Apr 02 2018