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A067359 Real part of (5+12i)^n. 3


%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(2n arctan(2/3)) or denominator of tan(2n 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.

%F G.f.: (1-5*x)/(1-10*x+169*x^2). a(n)=10*a(n-1)-169*a(n-2). - _Michael Somos_, Jun 27 2002

%F a(n)=(1/2)*[(5-12*I)^n+(5+12*I)^n], with n>=0 and I=sqrt(-1) - _Paolo P. Lava_, Jul 15 2008

%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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Last modified June 25 09:48 EDT 2019. Contains 324347 sequences. (Running on oeis4.)