A375896 - OEIS

%I #14 Sep 28 2024 07:38:48

%S 0,-1,-122,-10788,-816424,-55416080,-3416689056,-189851801152,

%T -9167161367168,-340760709275904,-4024113571740160,904814009441803264,

%U 126870078341747693568,11772031375019592445952,916527986864591725551616,63598173885399939858677760

%N a(n) = (64)^n*sin (nA - nB)/(5 sqrt(15)), where A, B, C are, respectively, the angles opposite sides BC, CA, AB in a triangle ABC having sidelengths  |BC| = 2, |CA| = 3, |AB| = 4; ABC is the smallest integer-sided scalene triangle.

%C If a prime p divides a term, then the indices n such that p divides a(n) comprise   an arithmetic sequence; see the Renault paper in References. For a guide to related sequences, see A375880.

%H Marc Renault, <a href="https://www.jstor.org/stable/10.4169/math.mag.86.5.372">The Period, Rank, and Order of the (a,b)-Fibonacci Sequence mod m</a>, Math. Mag. 86 (2013) pp. 372-380.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (122,-4096).

%F a(n) = (64)^n*cos (nA - nB)/(5 sqrt(15)), where A, B, C are, respectively, the angles opposite sides BC, CA, AB in a triangle ABC having sidelengths |BC| = 2, |CA| = 3, |AB| = 4.

%F a(n) = 122 a(n-1) - 4096 a(n-2), where a(0) = 0, a(1) = -1.

%F From _Stefano Spezia_, Sep 25 2024: (Start)

%F G.f.: -x/(1 - 122*x + 4096*x^2).

%F E.g.f.: -exp(61*x)*sin(5*sqrt(105)*x)/(5*sqrt(5)). (End)

%t (* Program 1 *)

%t A[a_, b_, c_] := ArcSin[Sqrt[1 - ((b^2 + c^2 - a^2)/(2  b  c))^2]];

%t {a, b, c} = {2, 3, 4};

%t Table[TrigExpand[(64)^n  Sin[n (A[a, b, c] - A[b, c, a])]/(5 Sqrt[15])], {n, 0, 18}]

%t (* Program 2 *)

%t LinearRecurrence[{122, -4096}, {0, -1}, 30]

%Y Cf. A375880.

%K sign,easy

%O 0,3

%A _Clark Kimberling_, Sep 25 2024