A066770 a(n) = 5^n*sin(2n*arctan(1/2)) or numerator of tan(2n*arctan(1/2)).

4, 24, 44, -336, -3116, -10296, 16124, 354144, 1721764, 1476984, -34182196, -242017776, -597551756, 2465133864, 29729597084, 116749235904, -42744511676, -3175197967656, -17982575014036, -28515500892816, 278471369994004, 2383715742284424, 7340510203856444

Offset: 1

Author

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

Keywords

• sign
• easy
• frac

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 (6,-25).

Crossrefs

Cf. A066771, A000351 powers of 5 and also hypotenuse of right triangle with legs given by A066770 and A066771.

Note that A066770, A066771 and A000351 are primitive Pythagorean triples with hypotenuse 5^n. The offset of A000351 is zero, but the offset is 1 for A066770, A066771.

Programs

Maple

a[1] := 4/3; for n from 1 to 40 do a[n+1] := (4/3+a[n])/(1-4/3*a[n]):od: seq(abs(numer(a[n])), n=1..40);# a[n]=tan(2n arctan(1/2))

Mathematica

Table[ 5^n*Sin[2*n*ArcCot[2]] // Simplify, {n, 1, 23}] (* Jean-François Alcover, Mar 04 2013 *)

PARI

a(n)=imag((2+I)^(2*n))

Formula

G.f.: 4*x/(1-6*x+25*x^2). - Ralf Stephan, Jun 12 2003

a(n) = 5^n*sin(2*n*arctan(1/2)). A recursive formula for T(n) = tan(2*n*arctan(1/2)) is T(n+1) = (4/3+T(n))/(1-4/3*T(n)). Unsigned a(n) is the absolute value of numerator of T(n).

a(n) is the imaginary part of (2+I)^(2*n) = Sum_{k=0..n} 2^(2*n-2*k-1)*(-1)^k*binomial(2*n, 2*k+1). - Benoit Cloitre, Aug 03 2002

a(n) = 6*a(n-1)-25*a(n-2), n>2. - Gary Detlefs, Dec 11 2010

a(n) = 5^n*sin(n*x), where x = arcsin(4/5) = 0.927295218.. . - Gary Detlefs, Dec 11 2010