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 A106709 #26 Sep 08 2022 08:45:18
%S A106709 0,-2,-10,-46,-210,-958,-4370,-19934,-90930,-414782,-1892050,-8630686,
%T A106709 -39369330,-179585278,-819187730,-3736768094,-17045465010,
%U A106709 -77753788862,-354678014290,-1617882493726,-7380056440050,-33664517212798,-153562473183890,-700483331493854
%N A106709 Expansion of g.f. -2*x/(1 - 5*x + 2*x^2).
%C A106709 Let T(n,k) denote the k-th element of row n of Stern's triangle (see A337277). Then b(n) = Sum_k T(n,k)*T(n,k+1) gives the present sequence (without the signs). - _N. J. A. Sloane_, Nov 19 2020
%H A106709 G. C. Greubel, <a href="/A106709/b106709.txt">Table of n, a(n) for n = 0..1000</a>
%H A106709 Richard P. Stanley, <a href="https://arxiv.org/abs/1901.04647">Some Linear Recurrences Motivated by Stern's Diatomic Array</a>, arXiv:1901.04647 [math.CO], 2019. Also American Mathematical Monthly 127.2 (2020): 99-111.
%H A106709 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-2).
%F A106709 a(n) = -2*A107839(n-1), n>0.
%F A106709 a(n) = first entry of v(n), where v(n) = M*v(n-1), M is the 2 X 2 matrix ({0, -2}, {1, 5}) and v(0) is the column vector (0, 1).
%F A106709 G.f.: -2*x/(1-5*x+2*x^2). - _Alois P. Heinz_, Nov 26 2020
%F A106709 a(n) = -sqrt(2)^(n+1)*ChebyshevU(n-1, 5/(2*sqrt(2))). - _G. C. Greubel_, Sep 10 2021
%p A106709 a:= n-> (<<0|-2>, <1|5>>^n)[1,2]:
%p A106709 seq(a(n), n=0..25); # _Alois P. Heinz_, Nov 19 2020
%t A106709 LinearRecurrence[{5,-2}, {0,-2}, 41] (* _G. C. Greubel_, Sep 10 2021 *)
%o A106709 (Magma) I:=[0,-2]; [n le 2 select I[n] else 5*Self(n-1) - 2*Self(n-2): n in [1..41]]; // _G. C. Greubel_, Sep 10 2021
%o A106709 (Sage) [-round(sqrt(2)^(n+1)*chebyshev_U(n-1, 5/(2*sqrt(2)))) for n in (0..40)] # _G. C. Greubel_, Sep 10 2021
%Y A106709 Cf. A107839, A337277.
%K A106709 sign,easy,less
%O A106709 0,2
%A A106709 _Roger L. Bagula_, May 30 2005
%E A106709 Edited by _N. J. A. Sloane_, Apr 30 2006
%E A106709 New name by _G. C. Greubel_, Sep 10 2021