A375885 - OEIS

%I #7 Sep 25 2024 11:49:00

%S 0,1,2,-12,-56,80,1056,832,-15232,-43776,156160,1012736,-473088,

%T -17149952,-26730496,220938240,869564416,-1795883008,-17504796672,

%U -6275465216,267525816320,635459076096,-3009494908928,-16186335035392,15779248472064,290539857510400

%N a(n) = 4^n sin(nC)/sqrt(15), where B is the angle opposite side CA 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 (2,-16).

%F a(n) = 2^(2n-1) sin(nA)/sqrt(15), where A is the angle opposite side BC in a triangle ABC having sidelengths |BC|=2, |CA| = 3, |AB| = 4  (the smallest integer-sided scalene triangle).

%F a(n) = 2 a(n-1) - 16 a(n-2), where a(0) = 0, a(1) = 1.

%F From _Stefano Spezia_, Sep 23 2024: (Start)

%F G.f.: x/(1 - 2*x + 16*x^2).

%F E.g.f.: exp(x)*sin(sqrt(15)*x)/sqrt(15). (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[4^n  Sin[n  A[c, a, b]]]/Sqrt[15], {n, 0, 30}]

%t (* Program 2 *)

%t LinearRecurrence[{2, -16}, {0, 1}, 30]

%Y Cf. A375880.

%K sign,easy

%O 0,3

%A _Clark Kimberling_, Sep 22 2024