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 A089817 #49 May 31 2025 05:18:18
%S A089817 1,6,30,145,696,3336,15985,76590,366966,1758241,8424240,40362960,
%T A089817 193390561,926589846,4439558670,21271203505,101916458856,488311090776,
%U A089817 2339638995025,11209883884350,53709780426726,257339018249281
%N A089817 a(n) = 5*a(n-1) - a(n-2) + 1 with a(0)=1, a(1)=6.
%C A089817 Partial sums of Chebyshev sequence S(n,5) = U(n,5/2) = A004254(n) (Chebyshev's polynomials of the second kind, see A049310). - _Wolfdieter Lang_, Aug 31 2004
%C A089817 In this sequence 4*a(n)*a(n+2)+1 is a square. - _Bruno Berselli_, Jun 19 2012
%H A089817 G. C. Greubel, <a href="/A089817/b089817.txt">Table of n, a(n) for n = 0..1000</a>
%H A089817 R. C. Alperin, <a href="https://www.fq.math.ca/Papers/58-2/alperin09212019.pdf">A nonlinear recurrence and its relations to Chebyshev polynomials</a>, Fib. Q., Vol. 58, No. 2 (2020), 140-142.
%H A089817 Ioana-Claudia Lazăr, <a href="https://arxiv.org/abs/1904.06555">Lucas sequences in t-uniform simplicial complexes</a>, arXiv:1904.06555 [math.GR], 2019.
%H A089817 F. M. van Lamoen, <a href="http://forumgeom.fau.edu/FG2006volume6/FG200637index.html">Square wreaths around hexagons</a>, Forum Geometricorum, 6 (2006) 311-325.
%H A089817 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A089817 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-6,1).
%F A089817 For n > 0, a(n-1) = Sum_{i=1..n} Sum_{j=1..i} b(n) with b(n) as in A004253.
%F A089817 a(n) = (2/3 - sqrt(21)/7)*(5/2 - sqrt(21)/2)^n + (2/3 + sqrt(21)/7)*(5/2 + sqrt(21)/2)^n - 1/3.
%F A089817 G.f.: 1/((1-x)*(1 - 5*x + x^2)) = 1/(1 - 6*x + 6*x^2 - x^3).
%F A089817 a(n) = 6*a(n-1) - 6*a(n-2) + a(n-3) for n >= 2, a(-1):=0, a(0)=1, a(1)=6.
%F A089817 a(n) = (S(n+1, 5) - S(n, 5) - 1)/3 for n >= 0.
%F A089817 a(n)*a(n-2) = a(n-1)*(a(n-1)-1) for n > 1. - _Bruno Berselli_, Nov 29 2016
%t A089817 Join[{a=1,b=6},Table[c=5*b-a+1;a=b;b=c,{n,60}]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 06 2011*)
%t A089817 CoefficientList[Series[1/(1 - 6*x + 6*x^2 - x^3), {x, 0, 50}], x] (* _G. C. Greubel_, Nov 20 2017 *)
%o A089817 (PARI) a(n)=([0,1,0; 0,0,1; 1,-6,6]^n*[1;6;30])[1,1] \\ _Charles R Greathouse IV_, Nov 29 2016
%o A089817 (PARI) x='x+O('x^50); Vec(1/(1-6*x+6*x^2-x^3)) \\ _G. C. Greubel_, Nov 20 2017
%o A089817 (Magma) [Round((2/3 - Sqrt(21)/7)*(5/2 - Sqrt(21)/2)^n + (2/3 + Sqrt(21)/7)*(5/2 + Sqrt(21)/2)^n - 1/3): n in [0..30]]; // _G. C. Greubel_, Nov 20 2017
%Y A089817 Cf. A061278, A053142, A101368.
%Y A089817 See. A212336 for more sequences with g.f. of the type 1/(1-k*x+k*x^2-x^3).
%K A089817 nonn,easy
%O A089817 0,2
%A A089817 _Paul Barry_, Nov 14 2003