This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A116470 #32 Jan 23 2025 08:31:52
%S A116470 0,1,2,3,4,5,7,8,11,13,18,21,29,34,47,55,76,89,123,144,199,233,322,
%T A116470 377,521,610,843,987,1364,1597,2207,2584,3571,4181,5778,6765,9349,
%U A116470 10946,15127,17711,24476,28657,39603,46368,64079,75025,103682,121393,167761
%N A116470 All distinct Fibonacci and Lucas numbers.
%C A116470 See A115339 for an essentially identical sequence.
%H A116470 Reinhard Zumkeller, <a href="/A116470/b116470.txt">Table of n, a(n) for n = 0..10000</a>
%H A116470 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,0,1).
%F A116470 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.
%F A116470 a(2*n) = Lucas(n+1) = Fibonacci(n) + Fibonacci(n+2) for n>1.
%F A116470 a(2*n+1) = Fibonacci(n+3) for n>2.
%F A116470 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
%F A116470 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
%p A116470 FL := (n, a, b) -> hypergeom([a - n/2, b - n/2], [1 - n], -4):
%p A116470 a := n -> `if`(n < 3, n, FL((n + 2 + 3*irem(n, 2))/2, irem(n, 2), 1/2)):
%p A116470 seq(simplify(a(n)), n=0..52); # _Peter Luschny_, Sep 03 2019
%t A116470 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 *)
%t A116470 With[{nn=50},Select[Union[Join[LucasL[Range[0,nn]],Fibonacci[Range[0,nn]]]],#<=200000&]] (* _Harvey P. Dale_, Jul 05 2019 *)
%o A116470 (Haskell)
%o A116470 import Data.List (transpose)
%o A116470 a116470 n = a116470_list !! n
%o A116470 a116470_list = 0 : 1 : 2 : concat
%o A116470 (transpose [drop 4 a000045_list, drop 3 a000032_list])
%o A116470 -- _Reinhard Zumkeller_, Aug 03 2013
%o A116470 (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
%o A116470 (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
%Y A116470 Union of A000045 and A000032.
%Y A116470 Cf. A288219 (even bisection).
%K A116470 nonn,easy
%O A116470 0,3
%A A116470 _Alexander Adamchuk_, Aug 13 2006