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

 

Logo

Annual Appeal: Today, Nov 11 2014, is the 4th anniversary of the launch of the new OEIS web site. 70,000 sequences have been added in these four years, all edited by volunteers. Please make a donation (tax deductible in the US) to help keep the OEIS running.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A132050 Denominator of 2*n*A000111(n-1)/A000111(n): approximations of Pi using Euler (up/down) numbers. 6
1, 1, 1, 5, 8, 61, 136, 1385, 3968, 50521, 176896, 2702765, 260096, 199360981, 951878656, 19391512145, 104932671488, 2404879675441, 14544442556416, 74074237647505, 2475749026562048, 69348874393137901, 507711943253426176 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

The rationals r(n)=2*n*e(n-1)/e(n), where e(n)=A000111(n), approximate Pi as n -> oo. - M. F. Hasler, Apr 03 2013

Numerators are given in A132049.

See the Delahaye reference and a link by W. Lang given in A132049.

From Paul Curtz, Mar 17 2013: (Start)

Apply the Akiyama-Tanigawa transform (or algorithm) to A046978(n+2)/A016116(n+1):

1,        1/2,      0,   -1/4,  -1/4,  -1/8,      0, 1/16, 1/16;

1/2,        1,    3/4,      0,  -5/8,  -3/4,  -7/16,    0;  = Balmer0(n)

-1/2,     1/2,    9/4,    5/2,   5/8, -15/8, -49/16;

-1,      -7/2,   -3/4,   15/2,  25/2,  57/8;

5/2,    -11/2,  -99/4,    -20, 215/8;

8,       77/2,  -57/4, -375/2;

-61/2,  211/2, 2079/4;

-136, -1657/2;

1385/2;

The first column is PIEULER(n) = 1, 1/2, -1/2, -1, 5/2, 8, -61/2, -136, 1385/2,... = c(n)/d(n). Abs c(n+1)=1,1,1,5,8,61,... =a(n) with offset=1.

For numerators of Balmer0(n) see A076109, A000265 and A061037(n-1) (End).

Other completely unrelated rational approximations of Pi are given by A063674/A063673 and other references there. - M. F. Hasler, Apr 03 2013

LINKS

Table of n, a(n) for n=1..23.

FORMULA

a(n)=denominator(r(n)) with the rationals r(n):=2*n*e(n-1)/e(n) where e(n):=A000111(n).

EXAMPLE

Rationals r(n): [2, 4, 3, 16/5, 25/8, 192/61, 427/136, 4352/1385, 12465/3968, 158720/50521,...].

MATHEMATICA

e[n_] := If[EvenQ[n], Abs[EulerE[n]], Abs[(2^(n + 1)*(2^(n + 1) - 1)*BernoulliB[n + 1])/(n + 1)]]; r[n_] := 2*n*(e[n - 1]/e[n]); a[n_] := Denominator[r[n]]; Table[a[n], {n, 1, 23}] (* Jean-Fran├žois Alcover, Mar 26 2013 *)

CROSSREFS

Cf. triangle A008281 (main diagonal give zig-zag numbers A000111).

Sequence in context: A117474 A165716 A068478 * A250067 A213239 A123819

Adjacent sequences:  A132047 A132048 A132049 * A132051 A132052 A132053

KEYWORD

nonn,frac,easy

AUTHOR

Wolfdieter Lang Sep 14 2007

EXTENSIONS

Definition made more explicit, and initial terms a(1)=a(2)=1 added by M. F. Hasler, Apr 03 2013

STATUS

approved

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

Content is available under The OEIS End-User License Agreement .

Last modified December 22 07:03 EST 2014. Contains 252328 sequences.