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A067358 Imaginary part of (5+12i)^n. 3

%I

%S 0,12,120,-828,-28560,-145668,3369960,58317492,13651680,-9719139348,

%T -99498527400,647549275812,23290743888720,123471611274972,

%U -2701419604443960,-47880898349909868,-22269070348069440,7869181117654073292,82455284065364468280,-505338768229893703548

%N Imaginary part of (5+12i)^n.

%C Also 13^n sin(2n arctan(2/3)) or numerator of tan(2n arctan(2/3)).

%C Note that a(n), A067359(n) and 13^n are primitive Pythagorean triples with hypotenuse 13^n.

%D Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 430-433.

%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.

%F G.f.: 12*x/(1-10*x+169*x^2). a(n)=10*a(n-1)-169*a(n-2). - _Michael Somos_, Jun 27 2002

%p a[1] := 12/5; for n from 1 to 40 do a[n+1] := (12/5+a[n])/(1-12/5*a[n]):od: seq(abs(numer(a[n])), n=1..40);# a[n]=tan(2n arctan(2/3))

%o (PARI) a(n)=imag((5+12*I)^n)

%Y Cf. A067359 (13^n cos(2n arctan(2/3))).

%Y Cf. A066770, A066771, A067360, A067361, A020888, A014498, A020892.

%K sign,easy,frac

%O 0,2

%A Barbara Haas Margolius, (b.margolius(AT)csuohio.edu), Jan 17 2002

%E Better description from _Michael Somos_, Jun 27 2002

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Last modified July 18 19:00 EDT 2019. Contains 325144 sequences. (Running on oeis4.)