A132049 Numerator of 2*n*A000111(n-1)/A000111(n): approximations of Pi, using Euler (up/down) numbers.

2, 4, 3, 16, 25, 192, 427, 4352, 12465, 158720, 555731, 8491008, 817115, 626311168, 2990414715, 60920233984, 329655706465, 7555152347136, 45692713833379, 232711080902656, 7777794952988025, 217865914337460224

Offset: 1

Author

Wolfdieter Lang Sep 14 2007

Keywords

• nonn
• frac
• easy

Comments

The denominators are given in A132050.

a(n)/n = 2, 2, 1, 4, 5, 32, 61, 544, ... are integers for n<=19. a(20)/20 = 58177770225664/5. - Paul Curtz, Mar 25 2013, Apr 04 2013

Examples

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

References

• J.-P. Delahaye, Pi - die Story (German translation), Birkhäuser, 1999 Basel, p. 31. French original: Le fascinant nombre Pi, Pour la Science, Paris, 1997.

Links

• Leonhard Euler, On the sums of series of reciprocals, (Presented to the St. Petersburg Academy on December 5, 1735), arXiv:math/0506415 [math.HO], 2005-2008.
• Wolfdieter Lang, Rationals and some values
• Wikipedia, Bernoulli number

Crossrefs

Cf. triangle A008281 (main diagonal give zig-zag numbers A000111), A223925.

Programs

Maple

S := proc(n, k) option remember;
if k=0 then `if`(n=0,1,0) else S(n,k-1)+S(n-1,n-k) fi end:
R := n -> 2*n*S(n-1,n-1)/S(n,n);
A132049 := n -> numer(R(n)); A132050 := n -> denom(R(n));
seq(A132049(i),i=3..22); # Peter Luschny, Aug 04 2011

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_] := Numerator[r[n]]; Table[a[n], {n, 3, 22}] (* Jean-François Alcover, Mar 18 2013 *)

Python

from itertools import islice, count, accumulate
from fractions import Fraction
def A132049_gen(): # generator of terms
    yield 2
    blist = (0,1)
    for n in count(2):
        yield Fraction(2*n*blist[-1],(blist:=tuple(accumulate(reversed(blist),initial=0)))[-1]).numerator
A132049_list = list(islice(A132049_gen(),40)) # Chai Wah Wu, Jun 09-11 2022

Formula

a(n)=numerator(r(n)) with the rationals r(n)=2*n*e(n-1)/e(n), where e(n)=A000111(n) ("zig-zag" or "up-down" numbers), i.e., e(2*k)=A000364(k) (Euler numbers, secant numbers, "zig"-numbers) and e(2*k+1)=A000182(k+1),k>=0, (tangent numbers, "zag"-numbers). Rationals in lowest terms.

Extensions

Entries confirmed by N. J. A. Sloane, May 10 2012

More explicit definition from M. F. Hasler, Apr 03 2013

a(1) and a(2) prepended by Paul Curtz, Apr 04 2013