OFFSET
0,1
COMMENTS
If a prime p divides a term, then the indices n such that p divides a(n) comprise an arithmetic sequence; e.g., 7 divides a(2n+1) for n >= 0; 17 divides a(4n+2) for n>= 0. See the Renault paper in References.
In the following guide to related sequences, each sequence has n-th term of the form (2^m*k)*f(U), where f is a trigonometric function and U is a vertex angle, A, B, or C, m is a positive integer dependent on n, and k is a real number invariant of n. Only f(U) appears in the guide.
cos(nA), this sequence
cos(nB), A375881
cos(nC), A375882
sin(nA), A375883
sin(nB), A375884
sin(nC), A375885
cos(nB-nC), A375891
cos(nC-nA), A375892
cos(nA-nB), A375893
sin(nB-nC), A375894
sin(nC-nA), A375895
sin(nA-nB), A375896
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 (7,-16).
FORMULA
a(n) = 2^(2 n + 1)*cos(nA), where A is 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) = 2, a(1) = 7.
From Stefano Spezia, Sep 12 2024: (Start)
G.f.: (2 - 7*x)/(1 - 7*x + 16*x^2).
E.g.f.: 2*exp(7*x/2)*cos(sqrt(15)*x/2). (End)
MATHEMATICA
(* Program 1 *)
A[a_, b_, c_] := ArcCos[(b^2 + c^2 - a^2)/(2 b c)];
Table[TrigExpand[2^(2n + 1) Cos[n A[2, 3, 4]]], {n, 0, 30}]
(* Program 2 *)
LinearRecurrence[{7, -16}, {2, 7}, 30]
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Clark Kimberling, Sep 11 2024
STATUS
approved