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 A215817 #45 Jun 02 2013 05:10:59 %S A215817 3,6,19,66,237,866,3202,11948,44917,169914,646134,2467988,9462498, %T A215817 36398004,140399901,542894726,2103745125,8167514346,31762430143, %U A215817 123704647562,482435457922,1883712663668,7363103647479,28809291337986,112820819490970,442175629583316 %N A215817 a(n) is the rational part of A(n) = (6-sqrt(7))*A(n-1) - (12-4*sqrt(7))*A(n-2) + (8-3*sqrt(7))*A(n-3) with A(0)=3, A(1)=6-sqrt(7), A(2)=19-4*sqrt(7). %C A215817 The Berndt-type sequence number 14 for the argument 2Pi/7 defined by requiring a(n) to be the rational part of the trigonometric sum A(n) := c(1)^(2*n) + c(2)^(2*n) + c(4)^(2*n), where c(j) := 2*cos(Pi/4 + 2*Pi*j/7) = 2*cos((7+8*j)*Pi/28). We note that (A(n)-a(n))/sqrt(7) = A215877(n) are all integers. We have A(n)=2^n*O(n;i/2), where O(n;d) denote the big omega function with index n for the argument d in C defined in comments to A215794 (see also Witula-Slota's paper - Section 6). From the respective recurrence relation for this function we generate the title recurrence for A(n). %H A215817 Roman Witula and Damian Slota, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Slota/witula13.html">New Ramanujan-Type Formulas and Quasi-Fibonacci Numbers of Order 7</a>, Journal of Integer Sequences, Vol. 10 (2007), Article 07.5.6 %H A215817 Roman Witula, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Witula/witula17.html">Ramanujan Type Trigonometric Formulas: The General Form for the Argument 2*Pi/7</a>, Journal of Integer Sequences, Vol. 12 (2009), Article 09.8.5 %F A215817 a(n) = rational part of c(1)^(2n) + c(2)^(2n) + c(4)^(2n) = (1-s(1))^n + (1-s(2))^n + (1-s(4))^n, where c(j) := 2*cos((7+8*j)/28) and s(j) := sin(2*Pi*j/7). %F A215817 Empirical g.f.: -(2*x-1)*(6*x^4 -40*x^3 +58*x^2 -24*x +3) / (x^6 -24*x^5 +86*x^4 -104*x^3 +53*x^2 -12*x +1). - _Colin Barker_, Jun 01 2013 %Y A215817 Cf. A215493, A215494, A215143, A215510, A094429, A215794. %K A215817 nonn %O A215817 0,1 %A A215817 _Roman Witula_, Aug 25 2012