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 A214778 #43 Oct 03 2019 04:08:42
%S A214778 3,3,21,84,381,1668,7374,32511,143445,632775,2791506,12314613,
%T A214778 54325650,239656134,1057236915,4663973199,20574997221,90766067772,
%U A214778 400412159841,1766407883376,7792462676946,34376247490935,151649926417857,668999726876127,2951274986626458
%N A214778 a(n) = 3*a(n-1) + 6*a(n-2) + a(n-3), with a(0) = 3, a(1) = 3, and a(2) = 21.
%C A214778 Ramanujan-type sequence number 3 for the argument 2Pi/9 is equal to the subsequence ax(3n) of the sequence ax(n), which (with its two conjugate sequences bx(n) and cx(n)) is defined in the comments to the sequence A214779 (we note that simultaneously we have bx(3n)=cx(3n)=0).
%C A214778 From example below follows that a(n) is equal to the sum of n-th powers of the roots of the polynomial x^3-3*x^2-6*x-1.
%C A214778 We note that all a(n) are divisible by 3 and a(n)/3 == 1 (mod 3). - _Roman Witula_, Oct 06 2012
%D A214778 R. Witula, E. Hetmaniok, D. Slota, Sums of the powers of any order roots taken from the roots of a given polynomial, Proceedings of the Fifteenth International Conference on Fibonacci Numbers and Their Applications, Eger, Hungary, 2012.
%H A214778 Roman Witula, <a href="https://doi.org/10.1515/dema-2013-0418">Ramanujan Type Trigonometric Formulae</a>, Demonstratio Math. 45 (2012) 779-796.
%H A214778 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,6,1).
%F A214778 a(n) = (c(1)/c(2))^n + (c(2)/c(4))^n + (c(4)/c(1))^n, where c(j) := Cos(2*Pi*j/9).
%F A214778 G.f.: (3-6*x-6*x^2)/(1-3*x -6*x^2-x^3).
%F A214778 a(n+1) = A214951(n+1) - A214951(n). - _Roman Witula_, Oct 06 2012
%e A214778 From a(1)=3 (after squaring) and a(2)=21 the following equality follows c(1)/c(4) + c(4)/c(2) + c(2)/c(1) = -6, which implies the decomposition x^3 - 3*x^2 - 6*x - 1 =(x - c(1)/c(2))*(x - c(2)/c(4))*(x - c(4)/c(1)).
%t A214778 LinearRecurrence[{3, 6, 1}, {3, 3, 21}, 40] (* _T. D. Noe_, Jul 30 2012 *)
%o A214778 (PARI) Vec((3-6*x-6*x^2)/(1-3*x -6*x^2-x^3)+O(x^99)) \\ _Charles R Greathouse IV_, Oct 01 2012
%o A214778 (PARI) polsym(x^3 - 3*x^2 - 6*x - 1, 30) \\ _Charles R Greathouse IV_, Jul 20 2016
%Y A214778 Cf. A214699, A214779.
%K A214778 nonn,easy
%O A214778 0,1
%A A214778 _Roman Witula_, Jul 28 2012