%I A066771
%S A066771 1,3,7,117,527,237,11753,76443,164833,922077,9653287,34867797,32125393,
%T A066771 1064447283,5583548873,6890111163,98248054847,761741108157,2114245277767,
%U A066771 6358056037323,91004468168113,387075408075603,47340744250793,9392840736385317
%V A066771 1,3,-7,-117,-527,-237,11753,76443,164833,-922077,-9653287,-34867797,32125393,
%W A066771 1064447283,5583548873,6890111163,-98248054847,-761741108157,-2114245277767,
%X A066771 6358056037323,91004468168113,387075408075603,47340744250793,-9392840736385317
%N A066771 5^n cos(2n arctan(1/2)) or denominator of tan(2n arctan(1/2)).
%C A066771 Let A =
%C A066771 [ -(3/5)-(2/5)i,-(2/5)i,-(2/5)i,-(2/5)i ]
%C A066771 [ -(2/5)i,-(3/5)+(2/5)i,-(2/5)i,(2/5)i ]
%C A066771 [ -(2/5)i,-(2/5)i,-(3/5)+(2/5)i,(2/5)i ]
%C A066771 [ -(2/5)i,(2/5)i,(2/5)i,-(3/5)-(2/5)i ]
%C A066771 be the Cayley transform of the matrix iH, where H =
%C A066771 [1,1,1,1]
%C A066771 [1,-1,1,-1]
%C A066771 [1,1,-1,-1]
%C A066771 [1,-1,-1,1]
%C A066771 is an 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 (ss54(AT)york.ac.uk), Apr 28 2004
%C A066771 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 (qntmpkt(AT)yahoo.com), 
               Aug 06 2006
%D A066771 J. M. Borwein and R. Girgensohn, Addition theorems and binary expansions, 
               Canadian J. Math. 47 (1995) 262-273.
%D A066771 E. Eckert, The group of primitive Pythagorean triangles, Mathematics 
               Magazine 57 (1984) 22-27.
%D A066771 S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 430-433.
%H A066771 S. R. Finch, <a href="http://algo.inria.fr/bsolve/constant/plff/plff.html">
               Plouffe's Constant</a>
%H A066771 S. Plouffe, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               The Computation of Certain Numbers Using a Ruler and Compass</a>, 
               J. Integer Seqs. Vol. 1 (1998), #98.1.3.
%F A066771 A(n)=5^n cos(2n arctan(1/2)). A recursive formula for T(n) = tan(2n arctan(1/
               2)) is T(n+1)=(4/3+T(n))/(1-4/3*T(n)). Unsigned A(n) is the absolute 
               value of denominator of T(n)
%F A066771 a(n) is the real part of (2+I)^(2n) = sum(k=0, n, 4^(n-k)*(-1)^k*C(2n, 
               2k) ) - Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 03 2002
%F A066771 a(n) = real part of (3 + 4i)^n. - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Aug 06 2006
%p A066771 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))
%o A066771 (PARI) a(n)=if(n<0,0,real((2+I)^(2*n))
%Y A066771 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 A066771 Note that A066770, A066771 and A0000351 are primitive Pythagorean triples 
               with hypotenuse 5^n. The offset of A0000351 is zero, but the offset 
               is 1 for A066770, A066771.
%Y A066771 Cf. A093378.
%Y A066771 Sequence in context: A014014 A015884 A156201 this_sequence A139159 A042329 
               A125956
%Y A066771 Adjacent sequences: A066768 A066769 A066770 this_sequence A066772 A066773 
               A066774
%K A066771 sign,easy,frac
%O A066771 0,2
%A A066771 Barbara Haas Margolius, (b.margolius(AT)csuohio.edu), Jan 17 2002