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A115339 a(2n-1)=F(n+1), a(2n)=L(n), where F(n) and L(n) are the Fibonacci and the Lucas sequences.

%I A115339 #29 Feb 16 2025 08:33:00
%S A115339 1,1,2,3,3,4,5,7,8,11,13,18,21,29,34,47,55,76,89,123,144,199,233,322,
%T A115339 377,521,610,843,987,1364,1597,2207,2584,3571,4181,5778,6765,9349,
%U A115339 10946,15127,17711,24476,28657,39603,46368,64079,75025,103682,121393,167761
%N A115339 a(2n-1)=F(n+1), a(2n)=L(n), where F(n) and L(n) are the Fibonacci and the Lucas sequences.
%C A115339 Alternate Fibonacci and Lucas sequence respecting their natural order.
%C A115339 See A116470 for an essentially identical sequence.
%C A115339 The ratio a(n+1)/a(n) increasingly approximates two constants: (5-sqrt(5))/2 (A094874) and (5+3*sqrt(5))/10 (A176015) according to whether n is odd or even. - _Davide Rotondo_, Oct 27 2024
%H A115339 Reinhard Zumkeller, <a href="/A115339/b115339.txt">Table of n, a(n) for n = 1..10000</a>
%H A115339 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FibonacciNumber.html">Fibonacci Number</a>
%H A115339 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LucasNumber.html">Lucas Number.</a>
%H A115339 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,0,1).
%F A115339 a(n+2) = a(n) + a(n-2).
%F A115339 G.f.: x*( -1-x-x^2-2*x^3 ) / ( -1+x^2+x^4 ). - _R. J. Mathar_, Mar 08 2011
%t A115339 f[n_] := If[OddQ@n, Fibonacci[(n + 3)/2], Fibonacci[n/2 - 1] + Fibonacci[n/2 + 1]]; Array[f, 50] (* _Robert G. Wilson v_, Apr 29 2006 *)
%o A115339 (Haskell)
%o A115339 a115339 n = a115339_list !! (n-1)
%o A115339 a115339_list = [1, 1, 2, 3] ++
%o A115339                zipWith (+) a115339_list (drop 2 a115339_list)
%o A115339 -- _Reinhard Zumkeller_, Aug 03 2013
%o A115339 (PARI) x='x+O('x^50); Vec(x*(-1-x-x^2-2*x^3)/(-1+x^2+x^4)) \\ _G. C. Greubel_, Apr 27 2017
%Y A115339 Cf. A000045, A000032.
%Y A115339 Cf. A000930.
%Y A115339 Cf. A094874, A176015.
%K A115339 easy,nonn
%O A115339 1,3
%A A115339 _Giuseppe Coppoletta_, Mar 06 2006
%E A115339 More terms from _Robert G. Wilson v_, Apr 29 2006