OFFSET
0,2
COMMENTS
sqrt (a(n)^2 + (A139031(n))^2) = 5^n. Example: a(3) = -44, A139031(3) = 117. Sqrt (-44^2 + 117^2) = 5^3.
If a prime p divides a term, then the indices n such that p divides a(n) comprise an arithmetic sequence; e.g., 11 divides a(6n+3) for n >= 0; 31 divides a(8n+4) for n>= 0. See the Renault paper in Links. - Clark Kimberling, Oct 02 2024
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
Marc Renault, The Period, Rank, and Order of the (a,b)-Fibonacci Sequence mod m, Math. Mag. 86 (2013) 372-380.
Index entries for linear recurrences with constant coefficients, signature (8,-25).
FORMULA
Real part of (4 + 3i)^n. Term (1,1) of [4,-3; 3,4]^n. a(n), n>=2 = 8*a(n-1) - 25*a(n-2), given a(0) = 1, a(1) = 4. Odd-indexed terms of A066770 interleaved with even-indexed terms of A066771, irrespective of sign.
G.f.: (1-4*x) / ( 1-8*x+25*x^2 ). - R. J. Mathar, Feb 05 2011
a(n) = 5^n * cos(nB-nC), where B is the angle opposite side CA and C is the angle opposite side AB in a triangle ABC having sidelengths |BC|=3, |CA|=4, |AB|=5; ABC is the smallest integer-sided right triangle. - Clark Kimberling, Oct 02 2024
E.g.f.: exp(4*x)*cos(3*x). - Stefano Spezia, Oct 03 2024
EXAMPLE
a(5) = -3116 since (4 + 3i)^5 = (-3116 - 237i) where -237 = A139031(5).
MAPLE
a:= n-> Re((4+3*I)^n):
seq(a(n), n=0..24); # Alois P. Heinz, Oct 15 2024
MATHEMATICA
Re[(4+3I)^Range[40]] (* or *) LinearRecurrence[{8, -25}, {4, 7}, 40] (* Harvey P. Dale, Nov 09 2011 *)
A[a_, b_, c_] := ArcCos[(b^2 + c^2 - a^2)/(2 b c)];
{a, b, c} = {3, 4, 5};
Table[TrigExpand[5^n Cos[n (A[b, c, a] - A[c, a, b])]], {n, 0, 50}] (* Clark Kimberling, Oct 02 2024 *)
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Gary W. Adamson, Apr 06 2008
EXTENSIONS
More terms from Harvey P. Dale, Nov 09 2011
a(0)=1 prepended by Alois P. Heinz, Oct 15 2024
STATUS
approved