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

 

Logo

Annual appeal: Please make a donation to keep the OEIS running! Over 6000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A089499 a(0)=0; a(1)=1; a(2n)=4*Sum_{k=0...n}a(2k-1); a(2n+1)=a(2n)+a(2n-1). 2
0, 1, 4, 5, 24, 29, 140, 169, 816, 985, 4756, 5741, 27720, 33461, 161564, 195025, 941664, 1136689, 5488420, 6625109, 31988856, 38613965, 186444716, 225058681, 1086679440, 1311738121, 6333631924, 7645370045, 36915112104, 44560482149 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

1, 4, 5, 24, 29, 140,...= numerators in convergents to (sqrt(8) - 2) = continued fraction [0; 1, 4, 1, 4, 1, 4,...]; where sqrt(8) - 2 = .828427124... = the inradius of a right triangle with hypotenuse 6, legs sqrt(32) and 2. Denominators of convergents to [0; 1, 4, 1, 4, 1, 4,...] = A041011 starting (1, 5, 6, 29, 35,...). - Gary W. Adamson, Dec 22 2007

This is a strong divisibility sequence, that is, GCD(a(n),a(m)) = a(GCD(n,m)) for all natural numbers n and m. - Peter Bala, May 12 2014

LINKS

Table of n, a(n) for n=0..29.

Index to divisibility sequences

Index entries for linear recurrences with constant coefficients, signature (0, 6, 0, -1).

FORMULA

For n>0, a(n)=A001333(n)+A084068(n-1)*(-1)^n; e.g. 29=41-12. a(n)*a(n+1)=A046729(n); cf. A001333. a(2n+1)=A001653(n); a(2n)=A005319(n).

a(1) = 1, a(2n) = 4*a(2n-1) + a(2n-2); a(2n-1) = a(2n-2) + a(2n-3). Given the 2 X 2 matrix X = [1, 4; 1, 5], [a(2n-1), a(2n)] = top row of X^n. The sequence starting (1, 4, 5, 24, 29,...) = numerators in continued fraction [0; 1, 4, 1, 4, 1, 4,...] = (sqrt(8) - 2) = .828427124... E.g. X^3 = [29, 140; 35, 169], where 29/35, 140/169 are convergents to (sqrt(8)-2). - Gary W. Adamson, Dec 22 2007

a(n)=A000129(n)*A000034(n+1). a(n)=6*a(n-2)-a(n-4). G.f.: -x*(-1-4*x+x^2)/((x^2-2*x-1)*(x^2+2*x-1)). - R. J. Mathar, Jul 08 2009

From Peter Bala, May 12 2014: (Start)

a(2*n + 1) = A041011(2*n + 1); a(2*n) = 4*A041011(2*n).

For n odd, a(n) = (alpha^n - beta^n)/(alpha - beta), and for n even, a(n) = 4*(alpha^n - beta^n)/(alpha^2 - beta^2), where alpha = 1 + sqrt(2) and beta = 1 - sqrt(2).

a(n) = product {j = 1..floor(n/2)} ( 4 + 4*cos^2(j*Pi/n) ) for n >= 1. (End)

MATHEMATICA

Numerator[NestList[(4/(4+#))&, 0, 60]] (* Vladimir Joseph Stephan Orlovsky, Apr 13 2010 *)

PROG

(PARI) a(n)=([0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1; -1, 0, 6, 0]^n*[0; 1; 4; 5])[1, 1] \\ Charles R Greathouse IV, Nov 13 2015

CROSSREFS

Cf. A041011.

Sequence in context: A039583 A042123 A041531 * A249060 A042601 A219515

Adjacent sequences:  A089496 A089497 A089498 * A089500 A089501 A089502

KEYWORD

nonn,easy

AUTHOR

Charlie Marion, Nov 11 2003

EXTENSIONS

Corrected by T. D. Noe, Nov 08 2006

Definition corrected by Jonathan Sondow, Jun 06 2014

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 November 19 03:53 EST 2017. Contains 294912 sequences.