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
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 (11,-64).
FORMULA
a(n) = 2^(2n-1) sin(nA)/(3 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) = 7 a(n-1) - 16 a(n-2), where a(0) = 1, a(1) = -1.
From Stefano Spezia, Sep 23 2024: (Start)
G.f.: x/(1 - 11*x + 64*x^2).
E.g.f.: 2*exp(11*x/2)*sin(3*sqrt(15)*x/2)/(3*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[2^(3 n + 1) Sin[n A[b, c, a]]]/ (3 Sqrt[15]), {n, 0, 30}]
(*Program 2*)
LinearRecurrence[{11, -64}, {0, 1}, 30]
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Clark Kimberling, Sep 22 2024
STATUS
approved