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 A035344 #46 May 29 2025 16:43:13
%S A035344 1,5,19,67,231,791,2703,9231,31519,107615,367423,1254463,4283007,
%T A035344 14623103,49926399,170459391,581984767,1987020287,6784111615,
%U A035344 23162405887,79081400319,270000789503,921840357375,3147359850495,10745758687231,36688315047935,125261742817279
%N A035344 Expansion of 1/((1 - x)*(1 - 4*x + 2 * x^2)).
%D A035344 S. Bilotta, E. Pergola, R. Pinzani, and S. Rinaldi, Recurrence Relations, Succession Rules, and the Positivity Problem, in Language and Automata Theory and Applications, 9th International Conference, LATA 2015, Nice, France, March 2-6, 2015, Proceedings, Pages 499-510, Lecture Notes Comp. Sci. Vol. 8977.
%H A035344 Andrew Howroyd, <a href="/A035344/b035344.txt">Table of n, a(n) for n = 0..1000</a>
%H A035344 Stefano Bilotta, Elisa Pergola, Renzo Pinzani, and Simone Rinaldi, <a href="http://arxiv.org/abs/1301.2967">Recurrence relations versus succession rules</a>, arXiv preprint arXiv:1301.2967 [cs.DM], 2013.
%H A035344 Sela Fried, <a href="https://arxiv.org/abs/2505.14196">Even-up words and their variants</a>, arXiv:2505.14196 [math.CO], 2025. See p. 8.
%H A035344 László Németh, <a href="http://arxiv.org/abs/1511.02067">Hyperbolic Pascal pyramid</a>, arXiv:1511.02067 [math.CO], 2015 (2nd line of Table 2 is 6*a(n-3)).
%H A035344 László Németh, <a href="https://arxiv.org/abs/1701.06022">Pascal pyramid in the space H^2 x R</a>, arXiv:1701.06022 [math.CO], 2017 (2nd line of Table is 2*a(n-3)).
%H A035344 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,-6,2).
%F A035344 a(n) = 2*A007052(n)-1. The sequence 0, 0, 1, 5, 19, ... is the binomial transform of the Pell numbers A000129, preceded by an additional 0. a(n) = (1 + 1/sqrt(2))(2 + sqrt(2))^n + (1 - 1/sqrt(2))(2 - sqrt(2))^n - 1. - _Paul Barry_, Jul 16 2003
%F A035344 a(-1)=0, a(0)=1, a(n) = 4*a(n-1) - 2*a(n-2) + 1. - _Miklos Kristof_, Mar 09 2005
%F A035344 E.g.f.: exp(2*x)*(2*cosh(sqrt(2)*x) + sqrt(2)*sinh(sqrt(2)*x)) - cosh(x) - sinh(x). - _Stefano Spezia_, May 20 2024
%p A035344 a[ -1]:=0:a[0]:=1:for n from 1 to 50 do a[n]:=4*a[n-1]-2*a[n-2]+1 od: seq(a[n],n=0..50); # after _Miklos Kristof_
%t A035344 Join[{a=1,b=5},Table[c=4*b-2*a+1;a=b;b=c,{n,60}]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 06 2011 *)
%t A035344 CoefficientList[Series[1/((1-x)(1-4x+2x^2)),{x,0,30}],x] (* or *) LinearRecurrence[ {5,-6,2},{1,5,19},30] (* _Harvey P. Dale_, Mar 28 2016 *)
%o A035344 (PARI) Vec(1/((1-x)*(1-4*x+2*x^2))+O(x^99)) \\ _Charles R Greathouse IV_, Sep 24 2012
%Y A035344 Partial sums of A007070.
%Y A035344 Cf. A000129, A007052.
%K A035344 nonn,easy
%O A035344 0,2
%A A035344 _N. J. A. Sloane_
%E A035344 a(23) onwards from _Andrew Howroyd_, Jan 28 2024