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A132049
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Numerators of rationals which approximate Pi.
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3
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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; internal format)
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OFFSET
| 3,1
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COMMENTS
| The denominators are given in A132050.
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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.
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LINKS
| W. Lang, Rationals and some values.
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]
Wikipedia, Bernoulli number
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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.
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EXAMPLE
| Rationals r(n): [3, 16/5, 25/8, 192/61, 427/136, 4352/1385, 12465/3968, 158720/50521,...].
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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
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CROSSREFS
| Cf. triangle A008281 (main diagonal give zig-zag numbers A000111).
Sequence in context: A101132 A153723 A091273 * A101405 A193367 A013196
Adjacent sequences: A132046 A132047 A132048 * A132050 A132051 A132052
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KEYWORD
| nonn,frac,easy
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Sep 14 2007
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