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 A189236 #25 Jun 02 2025 04:01:18
%S A189236 5,2,14,32,114,347,1142,3649,11826,38111,123139,397443,1283406,
%T A189236 4143479,13378435,43194542,139463234,450284986,1453839839,4694021537,
%U A189236 15155624819,48933074467,157990585613,510105367936,1646980994190,5317619734147
%N A189236 Expansion of (5-8*x-15*x^2+4*x^3+4*x^4)/(1-2*x-5*x^2+2*x^3+4*x^4+x^5).
%C A189236 (Start) Let U be the unit-primitive matrix (see [Jeffery])
%C A189236 U=U_(11,3)=
%C A189236 (0 0 0 1 0)
%C A189236 (0 0 1 0 1)
%C A189236 (0 1 0 1 1)
%C A189236 (1 0 1 1 1)
%C A189236 (0 1 1 1 1).
%C A189236 Then a(n)=Trace(U^n). (End)
%C A189236 Evidently one of a class of accelerator sequences for Catalan's constant based on traces of successive powers of a unit-primitive matrix U_(N,r) (0<r<floor(N/2)) and for which the closed-form expression for a(n) is derived from the eigenvalues of U_(N,r).
%H A189236 L. E. Jeffery, <a href="/wiki/User:L._Edson_Jeffery/Unit-Primitive_Matrices">Unit-primitive matrices</a>
%H A189236 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2, 5, -2, -4, -1).
%F A189236 G.f.: (5-8*x-15*x^2+4*x^3+4*x^4)/(1-2*x-5*x^2+2*x^3+4*x^4+x^5).
%F A189236 a(n)=2*a(n-1)+5*a(n-2)-2*a(n-3)-4*a(n-4)-a(n-5), {a(m)}={5,2,14,32,114}, m=0..4.
%F A189236 a(n)=Sum_{k=1..5} ((x_k)^3-2*(x_k))^n; x_k=2*(-1)^(k-1)*cos(k*Pi/11).
%F A189236 Series expansion of g.f. at x=infinity gives -A062883 and all but the first term of -A189235.
%t A189236 CoefficientList[Series[ (5-8x-15x^2+4x^3+4x^4)/ (1-2x-5x^2+2x^3+4x^4+x^5), {x,0,29}],x] (* _Harvey P. Dale_, Apr 19 2011 *)
%t A189236 LinearRecurrence[{2, 5, -2, -4, -1}, {5, 2, 14, 32, 114}, 30] (* _T. D. Noe_, Apr 19 2011 *)
%o A189236 (PARI) Vec((5-8*x-15*x^2+4*x^3+4*x^4)/(1-2*x-5*x^2+2*x^3+4*x^4+x^5)+O(x^99)) \\ _Charles R Greathouse IV_, Sep 25 2012
%Y A189236 Cf. A062883, A189234, A189235, A189237.
%K A189236 nonn,easy
%O A189236 0,1
%A A189236 _L. Edson Jeffery_, Apr 18 2011