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

0, 12, 120, -828, -28560, -145668, 3369960, 58317492, 13651680, -9719139348, -99498527400, 647549275812, 23290743888720, 123471611274972, -2701419604443960, -47880898349909868, -22269070348069440, 7869181117654073292, 82455284065364468280, -505338768229893703548

Offset: 0

Author

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

Keywords

• sign
• easy
• frac

Comments

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

Note that a(n), A067359(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. A067359 (13^n cos(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(numer(a[n])), n=1..40);# a[n]=tan(2n arctan(2/3))

Mathematica

Im[(5 + 12*I)^Range[0, 24]] (* or *)
LinearRecurrence[{10, -169}, {0, 12}, 25] (* Paolo Xausa, Apr 22 2024 *)

PARI

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

Formula

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

Extensions

Better description from Michael Somos, Jun 27 2002