A375883 - OEIS

%I #9 Oct 02 2024 03:47:36

%S 0,1,7,33,119,305,231,-3263,-26537,-133551,-510265,-1435039,-1881033,

%T 9793393,98650279,533857665,2158599191,6568471697,11441714823,

%U -25003543391,-358092240905,-2106588992079,-9016647090073,-29411105757247,-61611386859561,39297984099025

%N 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; 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 (7,-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) = 7*a(n-1) - 16*a(n-2), where a(0) = 1, a(1) = -1.

%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[2^(3 n + 1)  Sin[n  A[b, c, a]]]/(3 Sqrt[15]), {n, 0, 30}]

%t (* Program 2 *)

%t LinearRecurrence[{7, -16}, {0, 1}, 30]

%Y Cf. A375880.

%K sign

%O 0,3

%A _Clark Kimberling_, Sep 22 2024