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

1,1

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

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

Table of n, a(n) for n=1..22.

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

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.

Example

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A329050 A053124 A242500 * A229213 A356063 A071970

Adjacent sequences: A132046 A132047 A132048 * A132050 A132051 A132052

Keyword

nonn,frac,easy

Author

Wolfdieter Lang Sep 14 2007

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

Status

approved