%I #64 Oct 15 2024 10:31:44
%S 1,3,-7,-117,-527,-237,11753,76443,164833,-922077,-9653287,-34867797,
%T 32125393,1064447283,5583548873,6890111163,-98248054847,-761741108157,
%U -2114245277767,6358056037323,91004468168113,387075408075603,47340744250793,-9392840736385317
%N a(n) = 5^n*cos(2*n*arctan(1/2)) or denominator of tan(2*n*arctan(1/2)).
%C Let A =
%C [ -3/5 -(2/5)i, -(2/5)i, -(2/5)i, -(2/5)i ]
%C [ -(2/5)i, -3/5 +(2/5)i, -(2/5)i, (2/5)i ]
%C [ -(2/5)i, -(2/5)i, -3/5 +(2/5)i, (2/5)i ]
%C [ -(2/5)i, (2/5)i, (2/5)i, -3/5 -(2/5)i ]
%C be the Cayley transform of the matrix iH, where H =
%C [1, 1, 1, 1]
%C [1, -1, 1, -1]
%C [1, 1, -1, -1]
%C [1, -1, -1, 1]
%C is a Hadamard matrix of order 4 and i is the imaginary unit. Any diagonal entry of the matrix A^n is one of the four complex numbers (+ or -)(X/5^n)(+ or -)(Y/(5^n)i). Then a(n) is the X in [A^n]_(j,j), j=1,2,3,4. - _Simone Severini_, Apr 28 2004
%C Related to the (3,4,5) Pythagorean triple. Each unsigned term represents a leg in a Pythagorean triple in which the hypotenuse = 5^n. E.g., (3 + 4i)^3 = (-117 + 44i), considered as two legs of a triangle, hypotenuse = 125 = 5^3. - _Gary W. Adamson_, Aug 06 2006
%C a(n) = 5^n*cos(nC-nA), where C is the angle opposite side AB and A is the angle opposite side BC in a triangle ABC having sidelengths |BC|=3, |CA|=4, |AB|=5 - _Clark Kimberling_, Oct 02 2024
%C 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; 13 divides a(6n+3) for n>= 0. See the Renault paper in References. - _Clark Kimberling_, Oct 03 2024
%D Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 430-433.
%H Seiichi Manyama, <a href="/A066771/b066771.txt">Table of n, a(n) for n = 0..1000</a>
%H J. M. Borwein and R. Girgensohn, <a href="http://dx.doi.org/10.4153/CJM-1995-013-4">Addition theorems and binary expansions</a>, Canadian J. Math. 47 (1995) 262-273.
%H E. Eckert, <a href="http://www.jstor.org/stable/2690291">The group of primitive Pythagorean triangles</a>, Mathematics Magazine 57 (1984) 22-27.
%H Steven R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/constant/plff/plff.html">Plouffe's Constant</a> [Broken link]
%H Steven R. Finch, <a href="http://web.archive.org/web/20010624104257/http://www.mathsoft.com/asolve/constant/plff/plff.html">Plouffe's Constant</a> [From the Wayback machine]
%H Simon Plouffe, <a href="https://cs.uwaterloo.ca/journals/JIS/compass.html">The Computation of Certain Numbers Using a Ruler and Compass</a>, J. Integer Seqs. Vol. 1 (1998), #98.1.3.
%H Marc Renault, <a href="https://www.jstor.org/stable/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) pp. 372-380.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-25).
%F G.f.: ( 1-3*x ) / ( 1-6*x+25*x^2 ).
%F A recursive formula for T(n) = tan(2*n*arctan(1/2)) is T(n+1) = (4/3 + T(n))/(1 - (4/3)*T(n)). Unsigned A(n) is the absolute value of the denominator of T(n).
%F a(n) is the real part of (2+i)^(2n) = Sum_{k=0..n} 4^(n-k)*(-1)^k*C(2n, 2k). - _Benoit Cloitre_, Aug 03 2002
%F a(n) = real part of (3 + 4i)^n. - _Gary W. Adamson_, Aug 06 2006
%F a(n) = 6*a(n-1) - 25*a(n-2). - _Gary Detlefs_, Jun 10 2010
%F a(n) = 5^n*cos(n*arccos(3/5)). - _Gary Detlefs_, Dec 11 2010
%F a(n) = (-1)^n * hypergeom([1,-n,1/2-n],[1/2,1],-4). - _Gerry Martens_, Jul 28 2023
%p a[1] := 4/3; for n from 1 to 40 do a[n+1] := (4/3+a[n])/(1-4/3*a[n]):od: seq(abs(denom(a[n])), n=1..40);# a[n]=tan(2n arctan(1/2))
%t CoefficientList[Series[(1-3x)/(1-6x+25x^2),{x,0,30}],x] (* or *) LinearRecurrence[{6,-25},{1,3},30] (* _Harvey P. Dale_, Jul 16 2011 *)
%o (PARI) a(n)=real((2+I)^(2*n))
%Y Cf. A066770 5^n sin(2n arctan(1/2)), A000351 powers of 5 and also hypotenuse of right triangle with legs given by A066770 and A066771.
%Y Note that A066770, A066771 and A000351 are primitive Pythagorean triples with hypotenuse 5^n. The offset of A000351 is 0, but the offset is 1 for A066770, A066771.
%Y Cf. A093378.
%Y Cf. A193410, A121622.
%Y Cf. A139030.
%K sign,easy,frac
%O 0,2
%A Barbara Haas Margolius, (b.margolius(AT)csuohio.edu), Jan 17 2002