A376285 a(n) = (20)^n * cos(nA), where A is the angle opposite side BC in a triangle ABC having sidelengths |BC|=3, |CA|=4, |AB|=5; ABC is the smallest integer-sided right triangle.

1, 16, 112, -2816, -134912, -3190784, -48140288, -264175616, 10802495488, 451350102016, 10122205069312, 143370521411584, 538974657445888, -40101019526365184, -1498822487822041088, -31921911799759241216, -421972182463479283712, -734345118927640592384

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+3) 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..17.

Index entries for linear recurrences with constant coefficients, signature (32,-400).

Formula

a(n) = (20^n * cos(nA), where A is the angle opposite side BC in a triangle ABC having sidelengths |BC|=3, |CA|=4, |AB|=5; ABC is the smallest integer-sided right triangle.

a(n) = 32 a(n-1) - 400 a(n-2), where a(0) = 1, a(1) = 16.

From Stefano Spezia, Oct 03 2024: (Start)

G.f.: (1 - 16*x)/(1 - 32*x + 400*x^2).

E.g.f.: exp(16*x)*cos(12*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[3, 4, 5]]], {n, 0, 30}]
(*Program 2*)
LinearRecurrence[{32, -400}, {1, 16}, 30]

Crossrefs

Cf. A066771, A374880, A139030, A376455.

Sequence in context: A205964 A222113 A229531 * A253460 A081194 A121148

Adjacent sequences: A376282 A376283 A376284 * A376286 A376287 A376288

Keyword

sign,easy

Author

Clark Kimberling, Oct 03 2024

Status

approved