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 A067361 a(n) = 17^n*cos(2*n*arctan(1/4)) or denominator of tan(2*n*arctan(1/4)). 4
 15, 161, 495, -31679, -1093425, -23647519, -393425745, -4968639359, -35359140465, 375162560801, 21473668418415, 535788072480961, 9867752001506895, 141189807098209121, 1383913884510780975, 713562283940993281, -378544244105385903345 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Note that A067360(n), A067361(n) and 17^n are primitive Pythagorean triples with hypotenuse 17^n. REFERENCES Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 430-433. LINKS J. M. Borwein and R. Girgensohn, Addition theorems and binary expansions, Canadian J. Math. 47 (1995) 262-273. E. Eckert, The group of primitive Pythagorean triangles, Mathematics Magazine 57 (1984) 22-27. Steven R. Finch, Plouffe's Constant [Broken link] Steven R. Finch, Plouffe's Constant [From the Wayback machine] Simon Plouffe, The Computation of Certain Numbers Using a Ruler and Compass, J. Integer Seqs. Vol. 1 (1998), #98.1.3. FORMULA a(n) = 17^n*cos(2*n*arctan(1/4)). A recursive formula for T(n) = tan(2*n*arctan(1/4)) is T(n+1) = (8/15+T(n))/(1-8/15*T(n)). Unsigned a(n) is the absolute value of denominator of T(n). Conjectures from Colin Barker, Jul 25 2017: (Start) G.f.: x*(15 - 289*x) / (1 - 30*x + 289*x^2). a(n) = ((15 - 8*i)^n + (15 + 8*i)^n)/2 where i=sqrt(-1). a(n) = 30*a(n-1) - 289*a(n-2) for n>2. (End) MAPLE a[1] := 8/15; for n from 1 to 40 do a[n+1] := (8/15+a[n])/(1-8/15*a[n]):od: seq(abs(numer(a[n])), n=1..40); # a[n]=tan(2n arctan(1/4)) MATHEMATICA Table[t = Tan[2 n ArcTan[1/4]] // TrigToExp // Simplify; Sign[t] * Denominator[t], {n, 1, 17}] (* Jean-François Alcover, Jul 25 2017 *) CROSSREFS Cf. A067360 (17^n sin(2n arctan(1/4))). Cf. A066770, A066771, A067358, A067359, A020888, A014498, A020892. Sequence in context: A095685 A263514 A323292 * A014920 A081034 A279157 Adjacent sequences:  A067358 A067359 A067360 * A067362 A067363 A067364 KEYWORD sign,easy,frac,changed AUTHOR Barbara Haas Margolius, (b.margolius(AT)csuohio.edu), Jan 17 2002 STATUS approved

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Last modified April 23 07:51 EDT 2019. Contains 322381 sequences. (Running on oeis4.)