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 A098111 #35 Mar 17 2024 13:56:22
%S A098111 1,0,-5,-25,-100,-375,-1375,-5000,-18125,-65625,-237500,-859375,
%T A098111 -3109375,-11250000,-40703125,-147265625,-532812500,-1927734375,
%U A098111 -6974609375,-25234375000,-91298828125,-330322265625,-1195117187500,-4323974609375,-15644287109375,-56601562500000
%N A098111 Inverse binomial transform of A098149.
%C A098111 A030191(n) + 2*a(n) + A093129(n+2) = 4*A093129(n+1). - _Creighton Dement_, Oct 18 2004
%C A098111 From _Wolfdieter Lang_, Oct 02 2013: (Start)
%C A098111 These numbers a(n) and those of A030191(n) =: b(n), both interspersed with zeros, appear in the formula for nonnegative powers of the algebraic number rho(10) := 2*cos(pi/10) = phi*sqrt(3-phi), with the golden section phi, in terms of the power basis of the number field Q(rho(10)) of degree 4 (see A187360, n=10). In a (regular) decagon rho(10) is the length ratio of a smallest diagonal to the side. rho(10)^n = sum(A(n,k)*rho(10)^k, k=0..3), with A(2*k+1,0) = 0, A(2*k,0) = a(k), k >= 0; A(2*k,1) = 0, A(2*k+1,1) = a(k), k >= 0; A(2*k+1,2) = 0, k >= 0, A(0,2) = 0, A(2*k,2) = b(k-1), k >= 1; and A(2*k,3) = 0, k >= 0, A(1,3) = 0, A(2*k+1,3) = b(k-1), k >= 1. (End)
%H A098111 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-5).
%H A098111 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.
%F A098111 G.f.: (1-5x)/(1-5x+5x^2).
%F A098111 From _Wolfdieter Lang_, Oct 02 2013: (Start)
%F A098111 a(n) = b(n) - 5*b(n-1), n >= 0, with b(n) = A030191(n) = (sqrt(5))^n*S(n, sqrt(5)), with Chebyshev S-polynomials (see A049310).
%F A098111 a(n) = 5*(a(n-1) - a(n-2)), n >= 1, a(-1) = 1 = a(0). (End)
%e A098111 Powers of rho(10) in the Q(rho(10)) power basis for n = 5: rho(10)^5 = 0*1 + a(2)*rho(10) + 0*rho(10)^2 + b(1)*rho(10)^3 = -5*rho(10) + 5*rho(10)^3. - _Wolfdieter Lang_, Oct 02 2013
%t A098111 LinearRecurrence[{5,-5},{1,0},40] (* _Harvey P. Dale_, Dec 08 2015 *)
%Y A098111 Cf. A030191, A049310, A098149, A187360.
%K A098111 easy,sign
%O A098111 0,3
%A A098111 _Creighton Dement_, Sep 23 2004
%E A098111 More terms from _David Wasserman_, Jan 16 2008