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 A221719 #24 Jun 05 2025 09:56:02
%S A221719 0,2,6,15,34,74,157,328,678,1391,2838,5766,11677,23588,47554,95719,
%T A221719 192426,386450,775485,1555152,3117070,6245087,12507886,25044430,
%U A221719 50135229,100345484,200812362,401821143,803960098,1608434426,3217700893,6436748056,12875674422,25754873423,51515449734,103040126934,206095184221,412214526260,824468140690
%N A221719 a(n) = 3*2^n - Fibonacci(n+3) - 1.
%C A221719 Number of 3-strand braids with n crossings, divided by 2.
%D A221719 Paul K. Stockmeyer, Personal communication, Jan 12 2013
%H A221719 Colin Barker, <a href="/A221719/b221719.txt">Table of n, a(n) for n = 0..1000</a>
%H A221719 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4,-1,2).
%F A221719 From _Colin Barker_, Jun 07 2015: (Start)
%F A221719 a(n) = 4*a(n-1) - 4*a(n-2) - a(n-3) + 2*a(n-4) for n>3.
%F A221719 G.f.: x*(2-2*x-x^2) / ((1-x)*(1-2*x)*(1-x-x^2)). (End)
%F A221719 a(n) = -1 + 3*2^n + ( (2-sqrt(5))*((1-sqrt(5))/2)^n - (2+sqrt(5))*((1+sqrt(5))/2)^n )/sqrt(5). - _Colin Barker_, Nov 03 2016
%F A221719 From _G. C. Greubel_, Jun 05 2025: (Start)
%F A221719 a(n) = A083329(n+1) - A000045(n+3).
%F A221719 a(n) = A104004(n) - 1.
%F A221719 E.g.f.: 3*exp(2*x) - exp(x) - (2/sqrt(5))*exp(x/2)*( 2*sinh(sqrt(5)*x/2) + sqrt(5)*cosh(sqrt(5)*x/2) ). (End)
%t A221719 LinearRecurrence[{4,-4,-1,2},{0,2,6,15},40] (* _Harvey P. Dale_, Aug 25 2015 *)
%t A221719 A221719[n_]:= 3*2^n -Fibonacci[n+3] -1; (* _G. C. Greubel_, Jun 05 2025 *)
%o A221719 (PARI) concat(0, Vec(x*(x^2+2*x-2)/((x-1)*(2*x-1)*(x^2+x-1)) + O(x^100))) \\ _Colin Barker_, Jun 07 2015
%o A221719 (Magma)
%o A221719 A221719:= func< n | 3*2^n - Fibonacci(n+3) - 1 >; // _G. C. Greubel_, Jun 05 2025
%o A221719 (SageMath)
%o A221719 def A221719(n): return 3*2**n - fibonacci(n+3) - 1 # _G. C. Greubel_, Jun 05 2025
%Y A221719 Cf. A000045, A083329, A104004.
%K A221719 nonn,easy
%O A221719 0,2
%A A221719 _N. J. A. Sloane_, Jan 31 2013