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

0,2

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

Table of n, a(n) for n=0..19.

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).

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

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 *)

Prog

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

Crossrefs

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

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

Sequence in context: A056320 A056311 A009050 * A268634 A061506 A059155

Adjacent sequences: A067355 A067356 A067357 * A067359 A067360 A067361

Keyword

sign,easy,frac,changed

Author

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

Extensions

Better description from Michael Somos, Jun 27 2002

Status

approved