login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A066771 5^n cos(2n arctan(1/2)) or denominator of tan(2n arctan(1/2)). 12
1, 3, -7, -117, -527, -237, 11753, 76443, 164833, -922077, -9653287, -34867797, 32125393, 1064447283, 5583548873, 6890111163, -98248054847, -761741108157, -2114245277767, 6358056037323, 91004468168113, 387075408075603, 47340744250793, -9392840736385317 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Let A =

[ -(3/5)-(2/5)i,-(2/5)i,-(2/5)i,-(2/5)i ]

[ -(2/5)i,-(3/5)+(2/5)i,-(2/5)i,(2/5)i ]

[ -(2/5)i,-(2/5)i,-(3/5)+(2/5)i,(2/5)i ]

[ -(2/5)i,(2/5)i,(2/5)i,-(3/5)-(2/5)i ]

be the Cayley transform of the matrix iH, where H =

[1,1,1,1]

[1,-1,1,-1]

[1,1,-1,-1]

[1,-1,-1,1]

is an Hadamard matrix of order 4 and i is the imaginary unit. Any diagonal entry of the matrix A^n is one of the four complex numbers (+ or -)(X/5^n)(+ or -)(Y/(5^n)i). Then a(n) is the X in [A^n]_(j,j), j=1,2,3,4. - Simone Severini, Apr 28 2004

Related to the (3,4,5) Pythagorean triple. Each unsigned term represents a leg in a Pythagorean triple in which the hypotenuse = 5^n. E.g. (3 + 4i)^3 = (-117 + 44i), considered as two legs of a triangle, hypotenuse = 125 = 5^3. - Gary W. Adamson, Aug 06 2006

REFERENCES

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 430-433.

LINKS

Table of n, a(n) for n=0..23.

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.

S. R. Finch, Plouffe's Constant

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).

FORMULA

G.f.: ( 1-3*x ) / ( 1-6*x+25*x^2 ).

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

a(n) is the real part of (2+I)^(2n) = sum(k=0, n, 4^(n-k)*(-1)^k*C(2n, 2k) ). - Benoit Cloitre, Aug 03 2002

a(n) = real part of (3 + 4i)^n. - Gary W. Adamson, Aug 06 2006

a(n) = 6*a(n-1)-25*a(n-2). - Gary Detlefs, Jun 10 2010

a(n) = 5^n*cos(n*arccos(3/5)). - Gary Detlefs, Dec 11 2010

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(denom(a[n])), n=1..40); # a[n]=tan(2n arctan(1/2))

MATHEMATICA

CoefficientList[Series[(1-3x)/(1-6x+25x^2), {x, 0, 30}], x] (* or *) LinearRecurrence[{6, -25}, {1, 3}, 30] (* Harvey P. Dale, Jul 16 2011 *)

PROG

(PARI) a(n)=real((2+I)^(2*n))

CROSSREFS

Cf. A066770 5^n sin(2n arctan(1/2)), 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 0, but the offset is 1 for A066770, A066771.

Cf. A093378.

Cf. A193410, A121622.

Sequence in context: A015884 A224936 A156201 * A139159 A042329 A125956

Adjacent sequences:  A066768 A066769 A066770 * A066772 A066773 A066774

KEYWORD

sign,easy,frac,changed

AUTHOR

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

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified June 24 07:19 EDT 2017. Contains 288697 sequences.