The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A041007

(Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
Showing entries 1-10 | older changes

A041007

Denominators of continued fraction convergents to sqrt(6).
(history; published version)

#50 by N. J. A. Sloane at Sun Jun 03 12:05:57 EDT 2018

STATUS

proposed

approved

#49 by Rogério Serôdio at Tue Apr 10 10:30:01 EDT 2018

STATUS

editing

proposed

#48 by Rogério Serôdio at Tue Apr 10 10:29:53 EDT 2018

FORMULA

(7) Sum_{k=0..n} a(2*k+1)*(A142239(2*k) + A142239(2*(k+1))) = Sum_{k=0..n} a(3+4*k);

#47 by Rogério Serôdio at Mon Apr 02 08:56:53 EDT 2018

FORMULA

(8) Sum_{k=0..n} (a(2*k-1) + a(2*k+1))*A142239(2*k) = Sum_{k=0..n} A142239(3+4*k). (End)

STATUS

proposed

editing

Discussion

Mon Apr 09

18:13

OEIS Server: This sequence has not been edited or commented on for a week
yet is not proposed for review.  If it is ready for review, please
visit https://oeis.org/draft/A041007 and click the button that reads
"These changes are ready for review by an OEIS Editor."

Thanks.
  - The OEIS Server

#46 by Joerg Arndt at Mon Apr 02 04:53:51 EDT 2018

STATUS

editing

proposed

Discussion

Mon Apr 02

07:03

Rogério Serôdio: Sorry! That's a mistake. It should be "Sum_{k=0..n} a(3+4*k)"

#45 by Joerg Arndt at Mon Apr 02 04:53:24 EDT 2018

MAPLE

with (numtheory): seq( nthnumer(cfrac(sin(Pi/4)*tan(Pi/3), 25), i)-nthdenom(cfrac(sin(Pi/4)*tan(Pi/3), 25), i), i=1..24 ); # Zerinvary Lajos, Feb 10 2007

STATUS

proposed

editing

#44 by Jon E. Schoenfield at Sun Apr 01 21:52:36 EDT 2018

STATUS

editing

proposed

#43 by Jon E. Schoenfield at Sun Apr 01 21:50:30 EDT 2018

COMMENTS

Sqrtsqrt(6) = 4/2 + 4/9 + 4/(9*89) + 4/(89*881) + 4/(881*8721), ...; where sqrt(6) = 2.4494897427... and the sum of the first 5 terms of this series = 2.449489737... - Gary W. Adamson, Dec 21 2007

Sqrtsqrt(6) = 2 + continued fraction [2, 4, 2, 4, 2, 4, ...] = 4/2 + 4/9 + 4/(9*89) + 4/(89*881) + 4/(881*8721) + ... - Gary W. Adamson, Dec 21 2007

For n > 0, a(n) equals the permanent of the n X n tridiagonal matrix with the main diagonal alternating sequence [2, 4, 2, 4, ...] and 1's along the superdiagonal and the subdiagonal. - Rogério Serôdio, Apr 01 2018

FORMULA

Empirical Gg.f.: (1+2*x-x^2)/(1-10*x^2+x^4). [_- _Colin Barker_, Dec 31 2011]

Recurrence formula: a(n) = (3 + (-1)^n)*a(n-1) + a(n-2), a(0) = 1, a(1) = 2.

MATHEMATICA

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[6], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Mar 16 2011 *)

AUTHOR

N. J. A. Sloane.

STATUS

proposed

editing

Discussion

Sun Apr 01

21:52

Jon E. Schoenfield: @Rogério -- in

(7) Sum_{k=0..n} a(2*k+1)*(A142239(2*k) + A142239(2*(k+1))) = Sum_{k=0} a(3+4*k);

... is the "{k=0}" part correct?  If so, why not just simplify "Sum_{k=0} a(3+4*k)" to "a(3)"?

Thanks!

#42 by Altug Alkan at Sun Apr 01 18:03:43 EDT 2018

STATUS

editing

proposed

#41 by Altug Alkan at Sun Apr 01 17:58:28 EDT 2018

COMMENTS

For n > 0, a(n) equals the permanent of the n X n tridiagonal matrix with the main diagonal alternating sequence [2, 4, 2, 4, ...]and 1's along the superdiagonal and the subdiagonal. _- _Rogério Serôdio_, Apr 01 2018

FORMULA

~~~ From _Rogério Serôdio_, Apr 01 2018: (Start)

STATUS

proposed

editing

Discussion

Sun Apr 01

18:03

Altug Alkan: Dear Mr. Serôdio, this is the correct format of sign. If you have single comment, you should use like that ..the subdiagonal. - ~~~~ and if you have contribution like in formula section you should use From ~~~~: (Start). You should use exactly four times '~' like that '~~~~'. This will make your sign with correct date. Thanks, best regards.