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 A132049 Numerator of 2*n*A000111(n-1)/A000111(n): approximations of Pi, using Euler (up/down) numbers. 10
 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), 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), 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-François Alcover, Mar 18 2013 *) CROSSREFS Cf. triangle A008281 (main diagonal give zig-zag numbers A000111), A223925. Sequence in context: A225546 A053124 A242500 * 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 a(1) and a(2) prepended by Paul Curtz, Apr 04 2013 STATUS approved

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Last modified July 16 04:09 EDT 2019. Contains 325064 sequences. (Running on oeis4.)