A376283 a(n) = (40)^n * cos(nB), where B is the angle opposite side CA in a triangle ABC having sidelengths |BC|=3, |CA|=4, |AB|=5; ABC is the smallest integer-sided right triangle.

1, 24, -448, -59904, -2158592, -7766016, 3080978432, 160312590336, 2765438844928, -123759079981056, -10365137990975488, -299512095597133824, 2207640196898357248, 585186082406535266304, 24556707640476321185792, 242424234892406990831616

Offset

0,2

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(4n+2) for n >= 0; 17 divides a(8n+4) for n>= 0. See the Renault paper in References.

References

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

Links

Table of n, a(n) for n=0..15.

Index entries for linear recurrences with constant coefficients, signature (48,-1600).

Formula

a(n) = (40)^n * cos(nB), where B is the angle opposite side CA in a triangle ABC having sidelengths |BC|=3, |CA|=4, |AB|=5; ABC is the smallest integer-sided right triangle.

a(n) = 48 a(n-1) - 1600 a(n-2), where a(0) = 1, a(1) = 24.

From Stefano Spezia, Oct 03 2024: (Start)

G.f.: (1 - 24*x)/(1 - 48*x + 1600*x^2).

E.g.f.: exp(24*x)*cos(32*x). (End)

Mathematica

(*Program 1*)
A[a_, b_, c_] := ArcCos[(b^2 + c^2 - a^2)/(2  b  c)];
Table[TrigExpand[(20)^n  Cos[n  A[4, 5, 3]]], {n, 0, 30}]
(*Program 2*)
LinearRecurrence[{48, -1600}, {1, 24}, 30]

Crossrefs

Cf. A374880, A139030, A376285, A376455.

Sequence in context: A187625 A024303 A060195 * A114201 A263476 A263470

Adjacent sequences: A376279 A376280 A376282 * A376284 A376285 A376286

Keyword

sign,easy

Author

Clark Kimberling, Oct 02 2024

Status

approved