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 A133586 #25 Dec 10 2017 10:18:34
%S A133586 1,2,3,6,8,16,21,42,55,110,144,288,377,754,987,1974,2584,5168,6765,
%T A133586 13530,17711,35422,46368,92736,121393,242786,317811,635622,832040,
%U A133586 1664080,2178309,4356618,5702887,11405774,14930352,29860704,39088169,78176338,102334155
%N A133586 Expansion of x*(1+2*x)/( (x^2-x-1)*(x^2+x-1) ).
%C A133586 For n>1 A133585(n) + a(n) = A000032(n+1).
%H A133586 Colin Barker, <a href="/A133586/b133586.txt">Table of n, a(n) for n = 1..1000</a>
%H A133586 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,0,-1).
%F A133586 Equals A133080 * A133566 * A000045, where A133080 and A133566 are infinite lower triangular matrices and the Fibonacci sequence as a vector (previous definition).
%F A133586 For odd-indexed terms, a(n) = F(n+1). For even-indexed terms, a(n) = 2*a(n-1).
%F A133586 For n>1 A133585(n) + a(n) = A000032(n+1).
%F A133586 a(n) = A147600(n) + 2*A147600(n-1). - _R. J. Mathar_, Jun 20 2015
%F A133586 a(n) = (2^(-2-n)*((1-sqrt(5))^n*(-5+sqrt(5)) - (-1-sqrt(5))^n*(-3+sqrt(5)) - (-1+sqrt(5))^n*(3+sqrt(5)) + (1+sqrt(5))^n*(5+sqrt(5))))/sqrt(5). - _Colin Barker_, Mar 28 2016
%e A133586 a(5) = F(6) = 8.
%e A133586 a(6) = 2*a(5) = 2*8 = 16.
%p A133586 A133586aux := proc(n,k)
%p A133586 add(A133080(n,j)*A133566(j,k),j=k..n) ;
%p A133586 end proc:
%p A133586 A000045 := proc(n)
%p A133586 combinat[fibonacci](n) ;
%p A133586 end proc:
%p A133586 A133586 := proc(n)
%p A133586 add(A133586aux(n,j)*A000045(j),j=0..n) ;
%p A133586 end proc: # _R. J. Mathar_, Jun 20 2015
%t A133586 CoefficientList[Series[(1 + 2 x)/((x^2 - x - 1) (x^2 + x - 1)), {x, 0, 50}], x] (* _Vincenzo Librandi_, Jun 21 2015 *)
%t A133586 LinearRecurrence[{0,3,0,-1},{1,2,3,6},40] (* _Harvey P. Dale_, Dec 10 2017 *)
%o A133586 (PARI) {a(n) = if( n%2, fibonacci(n+1), 2*fibonacci(n))}; /* _Michael Somos_, Jun 20 2015 */
%o A133586 (PARI) Vec(x*(1+2*x)/((x^2-x-1)*(x^2+x-1)) + O(x^50)) \\ _Colin Barker_, Mar 28 2016
%Y A133586 Cf. A001906 (bisection), A025169 (bisection), A000032, A133586.
%K A133586 nonn,easy
%O A133586 1,2
%A A133586 _Gary W. Adamson_, Sep 18 2007
%E A133586 New definition and A-number in previous definition corrected by _R. J. Mathar_, Jun 20 2015