A375885 - OEIS

A375885

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.

0, 1, 2, -12, -56, 80, 1056, 832, -15232, -43776, 156160, 1012736, -473088, -17149952, -26730496, 220938240, 869564416, -1795883008, -17504796672, -6275465216, 267525816320, 635459076096, -3009494908928, -16186335035392, 15779248472064, 290539857510400

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

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.

LINKS

Table of n, a(n) for n=0..25.

Marc Renault, The Period, Rank, and Order of the (a,b)-Fibonacci Sequence mod m, Math. Mag. 86 (2013) pp. 372-380.

Index entries for linear recurrences with constant coefficients, signature (2,-16).

FORMULA

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).

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

From Stefano Spezia, Sep 23 2024: (Start)

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

E.g.f.: exp(x)*sin(sqrt(15)*x)/sqrt(15). (End)

MATHEMATICA

(* Program 1 *)

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

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

Table[TrigExpand[4^n Sin[n A[c, a, b]]]/Sqrt[15], {n, 0, 30}]

(* Program 2 *)

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

CROSSREFS