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%I #31 Jan 01 2024 11:43:54
%S 1,5,-119,-2035,-239,341525,3455641,-23161315,-815616479,-4241902555,
%T 95420159401,1671083125805,584824319281,-276564805068235,
%U -2864483360640839,18094618450123325,665043872449535041,3592448206424508485,-76467932379726337079,-1371803070683005304755
%N Real part of (5 + 12i)^n.
%C Also 13^n*cos(2*n*arctan(2/3)) or denominator of tan(2*n*arctan(2/3)).
%C Note that A067358(n), a(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.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (10, -169).
%F From _Michael Somos_, Jun 27 2002: (Start)
%F G.f.: (1-5*x)/(1-10*x+169*x^2).
%F a(n) = 10*a(n-1) - 169*a(n-2). (End)
%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(denom(a[n])), n=1..40);# a[n]=tan(2n arctan(2/3))
%t Table[Re[(5+12I)^n],{n,0,20}] (* _Harvey P. Dale_, Aug 24 2014 *)
%o (PARI) a(n)=real((5+12*I)^n)
%Y Cf. A067358 (13^n sin(2n arctan(2/3))).
%Y Cf. A066770, A066771, A067360, A067361, A020888, A014498, A020892.
%K sign,easy,frac
%O 1,2
%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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