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Real part of (4 + 3i)^n.
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%I #47 Oct 15 2024 12:04:41

%S 1,4,7,-44,-527,-3116,-11753,-16124,164833,1721764,9653287,34182196,

%T 32125393,-597551756,-5583548873,-29729597084,-98248054847,

%U -42744511676,2114245277767,17982575014036,91004468168113,278471369994004,-47340744250793,-7340510203856444,-57540563024581727

%N Real part of (4 + 3i)^n.

%C sqrt (a(n)^2 + (A139031(n))^2) = 5^n. Example: a(3) = -44, A139031(3) = 117. Sqrt (-44^2 + 117^2) = 5^3.

%C 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

%H Alois P. Heinz, <a href="/A139030/b139030.txt">Table of n, a(n) for n = 0..1000</a>

%H Marc Renault, <a href="https://doi.org/10.4169/math.mag.86.5.372">The Period, Rank, and Order of the (a,b)-Fibonacci Sequence mod m</a>, Math. Mag. 86 (2013) 372-380.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8,-25).

%F 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.

%F G.f.: (1-4*x) / ( 1-8*x+25*x^2 ). - _R. J. Mathar_, Feb 05 2011

%F 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

%F E.g.f.: exp(4*x)*cos(3*x). - _Stefano Spezia_, Oct 03 2024

%e a(5) = -3116 since (4 + 3i)^5 = (-3116 - 237i) where -237 = A139031(5).

%p a:= n-> Re((4+3*I)^n):

%p seq(a(n), n=0..24); # _Alois P. Heinz_, Oct 15 2024

%t Re[(4+3I)^Range[40]] (* or *) LinearRecurrence[{8,-25},{4,7},40] (* _Harvey P. Dale_, Nov 09 2011 *)

%t A[a_, b_, c_] := ArcCos[(b^2 + c^2 - a^2)/(2 b c)];

%t {a, b, c} = {3, 4, 5};

%t Table[TrigExpand[5^n Cos[n (A[b, c, a] - A[c, a, b])]], {n, 0, 50}] (* _Clark Kimberling_, Oct 02 2024 *)

%Y Cf. A139031, A066771, A066770, A374880, A376283, A376455.

%K sign,easy

%O 0,2

%A _Gary W. Adamson_, Apr 06 2008

%E More terms from _Harvey P. Dale_, Nov 09 2011

%E a(0)=1 prepended by _Alois P. Heinz_, Oct 15 2024