A215877 - OEIS

%I #15 Jun 02 2013 05:21:18

%S 0,-1,-4,-16,-64,-254,-1000,-3913,-15248,-59263,-229996,-892033,

%T -3459544,-13421784,-52104416,-202436819,-787231328,-3064347392,

%U -11940020992,-46569416006,-181808493296,-710442293743,-2778591945620,-10876271461745,-42606078512048

%N a(n) = (A(n) - A215817(n))/sqrt(7), where 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), and A(2)=19-4*sqrt(7).

%C The Berndt-type sequence number 15 for the argument 2Pi/7 defined by requiring sqrt(7)*a(n) to be the irrational 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).

%C We note that A(n)-sqrt(7)*a(n)= A215817(n). For more facts on A(n) - see comments to A215817.

%H 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 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 sqrt(7)*a(n) = to the irrational part of c(1)^(2*n) + c(2)^(2*n) + c(4)^(2*n) = (1-s(1))^n + (1-s(2))^n + (1-s(4))^n, where c(j) = 2*cos((7+8*j)*Pi/28) and s(j) := sin(2*Pi*j/7).

%F Empirical g.f.: -x * (2*x-1)^2 * (x^2-4*x+1) / (x^6 -24*x^5 +86*x^4 -104*x^3 +53*x^2 -12*x +1). - _Colin Barker_, Jun 01 2013

%e We have a(2)/a(1) = a(3)/a(2) = a(4)/a(3) = 4, but a(5)-4*a(4)=2 and a(6)=4*(a(5)-a(2)). Moreover it follows

%e the relations: 4*A(1)-A(2) = 5 = (3+s(1))*(1-s(1)) + (3+s(2))*(1-s(2)) + (3+s(4))*(1-s(4)), 4*A(2)-A(3) = 10 =  (3+s(1))*(1-s(1))^2 + (3+s(2))*(1-s(2))^2 + (3+s(4))*(1-s(4))^2, 4*A(3)-A(4) = 27 = (3+s(1))*(1-s(1))^3 + (3+s(2))*(1-s(2))^3 + (3+s(4))*(1-s(4))^3, whereas 4*A(4)-a(5) = 82-2*sqrt(7) = (3+s(1))*(1-s(1))^4 + (3+s(2))*(1-s(2))^4 + (3+s(4))*(1-s(4))^4.

%Y Cf. A215493, A215494, A215143, A215510, A094429, A215817.

%K sign

%O 0,3

%A _Roman Witula_, Aug 25 2012