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 A215665 #26 Aug 07 2025 04:35:50
%S A215665 0,-3,-3,-9,-6,-24,-9,-66,-3,-189,57,-564,360,-1749,1644,-5607,6681,
%T A215665 -18465,25650,-62076,95415,-211878,348321,-731049,1256841,-2541468,
%U A215665 4501572,-8881245,16046184,-31145307,57019797,-109482105,202204698,-385466112,716096199
%N A215665 a(n) = 3*a(n-2) - a(n-3), with a(0)=0, a(1)=a(2)=-3.
%C A215665 The Berndt-type sequence number 6 for the argument 2Pi/9 defined by the first relation from the section "Formula" below. Two sequences connected with a(n) (possessing the respective numbers 5 and 7) are discussed in A215664 and A215666 - for more details see comments to A215664 and Witula's reference. We have a(n) - a(n+1) = A215664(n).
%C A215665 From initial values and the recurrence formula we deduce that a(n)/3 are all integers.
%C A215665 We note that a(10) is the first element of a(n) which is positive integer and all (-1)^n*a(n+10) are positive integer, which can be obtained from the title recurrence relation.
%C A215665 The following decomposition holds (X - c(1)*c(2)^n)*(X - c(2)*c(4)^n)*(X - c(4)*c(1)^n) = X^3 - a(n)*X^2 - A215917(n-1)*X + (-1)^n.
%C A215665 If X(n) = 3*X(n-2) - X(n-3), n in Z, with X(n) = a(n) for every n=0,1,..., then X(-n) = abs(A215919(n)) = (-1)^n*A215919(n) for every n=0,1,...
%D A215665 R. Witula, Ramanujan type formulas for arguments 2Pi/7 and 2Pi/9, Demonstratio Math., (in press, 2012).
%D A215665 D. Chmiela and R. Witula, Two parametric quasi-Fibonacci numbers of the nine order, (submitted, 2012).
%H A215665 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,-1).
%F A215665 a(n) = c(1)*c(2)^n + c(2)*c(4)^n + c(4)*c(1)^n, where c(j) := 2*cos(2*Pi*j/9).
%F A215665 G.f.: -3*x*(1+x)/(1-3*x^2+x^3).
%e A215665 We have a(1)=a(2)=a(8)=-3, a(3)=a(6)=-9, a(4)+a(11)=-10*a(10), and 47*a(5)=2*a(11).
%t A215665 LinearRecurrence[{0,3,-1}, {0,-3,-3}, 50]
%o A215665 (PARI) concat(0,Vec(-3*(1+x)/(1-3*x^2+x^3)+O(x^99))) \\ _Charles R Greathouse IV_, Oct 01 2012
%Y A215665 Cf. A215455, A215634, A215635, A215636, A215664, A214699, A215007, A214683.
%K A215665 sign,easy
%O A215665 0,2
%A A215665 _Roman Witula_, Aug 20 2012