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

1, 5, -119, -2035, -239, 341525, 3455641, -23161315, -815616479, -4241902555, 95420159401, 1671083125805, 584824319281, -276564805068235, -2864483360640839, 18094618450123325, 665043872449535041, 3592448206424508485, -76467932379726337079, -1371803070683005304755

Offset: 1

Author

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

Keywords

• sign
• easy
• frac

Comments

Also 13^n*cos(2*n*arctan(2/3)) or denominator of tan(2*n*arctan(2/3)).

Note that A067358(n), a(n) and 13^n are primitive Pythagorean triples with hypotenuse 13^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.
• Index entries for linear recurrences with constant coefficients, signature (10, -169).

Crossrefs

Cf. A067358 (13^n sin(2n arctan(2/3))).

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

Programs

Maple

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))

Mathematica

Table[Re[(5+12I)^n],{n,0,20}] (* Harvey P. Dale, Aug 24 2014 *)

PARI

a(n)=real((5+12*I)^n)

Formula

From Michael Somos, Jun 27 2002: (Start)

G.f.: (1-5*x)/(1-10*x+169*x^2).

a(n) = 10*a(n-1) - 169*a(n-2). (End)

Extensions

Better description from Michael Somos, Jun 27 2002