A132050 Denominator of 2*n*A000111(n-1)/A000111(n): approximations of Pi using Euler (up/down) numbers.

1, 1, 1, 5, 8, 61, 136, 1385, 3968, 50521, 176896, 2702765, 260096, 199360981, 951878656, 19391512145, 104932671488, 2404879675441, 14544442556416, 74074237647505, 2475749026562048, 69348874393137901, 507711943253426176

Offset: 1

Author

Wolfdieter Lang, Sep 14 2007

Keywords

• nonn
• frac
• easy

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

Examples

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

Crossrefs

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

Programs

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-François Alcover, Mar 26 2013 *)

Python

from itertools import count, islice, accumulate
from fractions import Fraction
def A132050_gen(): # generator of terms
    yield 1
    blist = (0,1)
    for n in count(2):
        yield Fraction(2*n*blist[-1],(blist:=tuple(accumulate(reversed(blist),initial=0)))[-1]).denominator
A132050_list = list(islice(A132050_gen(),40)) # Chai Wah Wu, Jun 09-11 2022

Formula

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

Extensions

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