A319512 - OEIS

%I #56 Oct 08 2023 15:00:31

%S 1,3,11,42,161,616,2352,8967,34153,129997,494606,1881355,7154980,

%T 27208132,103456689,393367835,1495638123,5686513994,21620239081,

%U 82199944512,312521862408,1188195487255,4517461948657,17175149855885,65298950120782,248262786503683

%N a(n) = 7*a(n-1) - 14*a(n-2) + 7*a(n-3), a(0) = 1, a(1) = 3, a(2) = 11.

%C Let {X,Y,Z} be the roots of the cubic equation

%C   t^3 + at^2 + bt + c = 0

%C where {a, b, c} are integers. Let {u, v, w} be three numbers such that {u + v + w, u*X + v*Y + w*Z, u*X^2 + v*Y^2 + w*Z^2} are integers. Then

%C {p(n) = u*X^n + v*Y^n + w*Z^n | n = 0, 1, 2, ...}

%C is an integer sequence with the recurrence relation:

%C p(n) = -a*p(n-1) - b*p(n-2) - c*p(n-3).

%C This sequence has (a, b, c) = (-7, 14, -7), (u, v, w) = (1/(sqrt(7)*tan(4*(Pi/7))), 1/(sqrt(7)*tan(8*(Pi/7))), 1/(sqrt(7)*tan(2*(Pi/7)))).

%H Colin Barker, <a href="/A319512/b319512.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,-14,7)

%F (X, Y, Z) = (4*sin^2(2*(Pi/7)), 4*sin^2(4*(Pi/7)), 4*sin^2(8*(Pi/7)));

%F a(n) = 7*a(n-1) - 14*a(n-2) + 7*a(n-3), a(0) = 1, a(1) = 3, a(2) = 11.

%F G.f.: (1 - 2*x)^2 / (1 - 7*x + 14*x^2 - 7*x^3). - _Colin Barker_, Dec 11 2018

%t LinearRecurrence[{7, -14, 7}, {1, 3, 11}, 30] (* _Amiram Eldar_, Dec 10 2018 *)

%t CoefficientList[Series[(1-2x)^2/(1-7x+14x^2-7x^3),{x,0,30}],x] (* _Harvey P. Dale_, Oct 08 2023 *)

%o (PARI) Vec((1 - 2*x)^2 / (1 - 7*x + 14*x^2 - 7*x^3) + O(x^40)) \\ _Colin Barker_, Dec 11 2018

%Y Cf. A215007, A215008, A215143, A215493, A215494, A215510, A217274, A217444, A094430.

%K nonn,easy

%O 0,2

%A _Kai Wang_, Dec 10 2018

%E More terms from _Felix Fröhlich_, Dec 10 2018