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A132049
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Numerator of 2*n*A000111(n-1)/A000111(n): approximations of Pi, using Euler (up/down) numbers.
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10
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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)
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OFFSET
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1,1
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COMMENTS
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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
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REFERENCES
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J.-P. Delahaye, Pi - die Story (German translation), Birkhäuser, 1999 Basel, p. 31. French original: Le fascinant nombre Pi, Pour la Science, Paris, 1997.
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LINKS
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FORMULA
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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
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Rationals r(n): [3, 16/5, 25/8, 192/61, 427/136, 4352/1385, 12465/3968, 158720/50521, ...].
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MAPLE
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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);
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,frac,easy
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AUTHOR
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EXTENSIONS
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a(1) and a(2) prepended by Paul Curtz, Apr 04 2013
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STATUS
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approved
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