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A132049 Numerator of 2*n*A000111(n-1)/A000111(n): approximations of Pi, using Euler (up/down) numbers. 8
2, 4, 3, 16, 25, 192, 427, 4352, 12465, 158720, 555731, 8491008, 817115, 626311168, 2990414715, 60920233984, 329655706465, 7555152347136, 45692713833379, 232711080902656, 7777794952988025, 217865914337460224 (list; graph; refs; listen; history; text; internal format)
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), Birkhaeuser, 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), last paragraph, arXiv:math/0506415v2 [math.HO]. [Peter Luschny, Nov 18 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-Fran├žois Alcover, Mar 18 2013 *)

CROSSREFS

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

Sequence in context: A183169 A225546 A053124 * A229213 A071970 A182103

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

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

STATUS

approved

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Last modified April 16 21:07 EDT 2014. Contains 240627 sequences.