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%I #45 Jun 11 2022 11:43:14
%S 2,4,3,16,25,192,427,4352,12465,158720,555731,8491008,817115,
%T 626311168,2990414715,60920233984,329655706465,7555152347136,
%U 45692713833379,232711080902656,7777794952988025,217865914337460224
%N Numerator of 2*n*A000111(n-1)/A000111(n): approximations of Pi, using Euler (up/down) numbers.
%C The denominators are given in A132050.
%C 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
%D J.-P. Delahaye, Pi - die Story (German translation), Birkhäuser, 1999 Basel, p. 31. French original: Le fascinant nombre Pi, Pour la Science, Paris, 1997.
%H Leonhard Euler, <a href="http://arxiv.org/abs/math/0506415">On the sums of series of reciprocals</a>, (Presented to the St. Petersburg Academy on December 5, 1735), arXiv:math/0506415 [math.HO], 2005-2008.
%H Wolfdieter Lang, <a href="/A132049/a132049.pdf">Rationals and some values</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Bernoulli_number">Bernoulli number</a>
%F 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.
%e Rationals r(n): [3, 16/5, 25/8, 192/61, 427/136, 4352/1385, 12465/3968, 158720/50521, ...].
%p S := proc(n, k) option remember;
%p if k=0 then `if`(n=0,1,0) else S(n,k-1)+S(n-1,n-k) fi end:
%p R := n -> 2*n*S(n-1,n-1)/S(n,n);
%p A132049 := n -> numer(R(n)); A132050 := n -> denom(R(n));
%p seq(A132049(i),i=3..22); # _Peter Luschny_, Aug 04 2011
%t 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 *)
%o (Python)
%o from itertools import islice, count, accumulate
%o from fractions import Fraction
%o def A132049_gen(): # generator of terms
%o yield 2
%o blist = (0,1)
%o for n in count(2):
%o yield Fraction(2*n*blist[-1],(blist:=tuple(accumulate(reversed(blist),initial=0)))[-1]).numerator
%o A132049_list = list(islice(A132049_gen(),40)) # _Chai Wah Wu_, Jun 09-11 2022
%Y Cf. triangle A008281 (main diagonal give zig-zag numbers A000111), A223925.
%K nonn,frac,easy
%O 1,1
%A _Wolfdieter Lang_ Sep 14 2007
%E Entries confirmed by _N. J. A. Sloane_, May 10 2012
%E More explicit definition from _M. F. Hasler_, Apr 03 2013
%E a(1) and a(2) prepended by _Paul Curtz_, Apr 04 2013
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