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 A094649 #60 Nov 22 2023 12:07:52
%S A094649 4,1,7,4,19,16,58,64,187,247,622,925,2110,3394,7252,12289,25147,44116,
%T A094649 87727,157492,307294,560200,1079371,1987891,3798310,7043041,13382818,
%U A094649 24927430,47191492,88165105,166501903,311686804,587670811,1101562312
%N A094649 An accelerator sequence for Catalan's constant.
%C A094649 From _L. Edson Jeffery_, Apr 03 2011: (Start)
%C A094649 Let U be the unit-primitive matrix (see [Jeffery])
%C A094649 U = U_(9,1) =
%C A094649 (0 1 0 0)
%C A094649 (1 0 1 0)
%C A094649 (0 1 0 1)
%C A094649 (0 0 1 1).
%C A094649 Then a(n) = Trace(U^n). (End)
%C A094649 a(n)==1 (mod 3), a(3*n+1)==1 (mod 9). - _Roman Witula_, Sep 14 2012
%H A094649 Michael De Vlieger, <a href="/A094649/b094649.txt">Table of n, a(n) for n = 0..3649</a>
%H A094649 A. Akbary and Q. Wang, <a href="http://dx.doi.org/10.1155/IJMMS.2005.2631">On some permutation polynomials over finite fields</a>, International Journal of Mathematics and Mathematical Sciences, 2005:16 (2005) 2631-2640.
%H A094649 A. Akbary and Q. Wang, <a href="http://dx.doi.org/10.1090/S0002-9939-05-08220-1">A generalized Lucas sequence and permutation binomials</a>, Proceeding of the American Mathematical Society, 134 (1) (2006), 15-22, sequence a(n) with l=9.
%H A094649 David M. Bradley, <a href="http://dx.doi.org/10.1023/A:1006945407723">A Class of Series Acceleration Formulae for Catalan's Constant</a>, The Ramanujan Journal, Vol. 3, Issue 2, 1999, pp. 159-173.
%H A094649 David M. Bradley, <a href="http://arxiv.org/abs/0706.0356">A Class of Series Acceleration Formulae for Catalan's Constant</a>, arXiv:0706.0356 [math.CA], 2007.
%H A094649 Russell A. Gordon, <a href="http://math.colgate.edu/~integers/x84/x84.pdf">Lucas Type Sequences and Sums of Binomial Coefficients</a>, Integers (2023) Vol 23, Art. No. A84. See p. 21.
%H A094649 L. E. Jeffery, <a href="/wiki/User:L._Edson_Jeffery/Unit-Primitive_Matrices">Unit-primitive matrices</a>
%H A094649 Genki Shibukawa, <a href="https://arxiv.org/abs/1907.00334">New identities for some symmetric polynomials and their applications</a>, arXiv:1907.00334 [math.CA], 2019.
%H A094649 Q. Wang, <a href="https://www.semanticscholar.org/paper/On-generalized-Lucas-sequences-Wang-Akbari/7e33b3b79703dc6790fca133e8c92cc0cafcfe4a">On generalized Lucas sequences</a>, Contemp. Math. 531 (2010) 127-141, Table 2 (k=4)
%H A094649 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-2,-1).
%F A094649 G.f.: ( 4-3*x-6*x^2+2*x^3 ) / ( (x-1)*(x^3+3*x^2-1) )
%F A094649 a(n) = 1+(2*cos(Pi/9))^n+(-2*sin(Pi/18))^n+(-2*cos(2*Pi/9))^n.
%F A094649 a(n) = 2^n*Sum_{k=1..4} cos((2*k-1)*Pi/9)^n. - _L. Edson Jeffery_, Apr 03 2011
%F A094649 a(n) = 1 + (-1)^n*A215664(n), which is compatible with the last two formulas above. - _Roman Witula_, Sep 14 2012
%F A094649 a(n) = 3*a(n-2) + a(n-3) - 3, with a(0)=4, a(1)=1, and a(2)=7. - _Roman Witula_, Sep 14 2012
%e A094649 We have a(0)+a(3)=a(1)+a(2)=8, a(3)+a(4)=a(2)+a(5)=23, and a(7)+a(8)=a(9)+a(3)=247. - _Roman Witula_, Sep 14 2012
%t A094649 LinearRecurrence[{1, 3, -2, -1}, {4, 1, 7, 4}, 34] (* _Jean-François Alcover_, Sep 21 2017 *)
%o A094649 (PARI) Vec((4-3*x-6*x^2+2*x^3)/(1-x-3*x^2+2*x^3+x^4)+O(x^66)) /* _Joerg Arndt_, Apr 08 2011 */
%Y A094649 Cf. A000032, A094648, A094650.
%K A094649 easy,nonn
%O A094649 0,1
%A A094649 _Paul Barry_, May 18 2004