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A002485
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Numerators of convergents to Pi.
(Formerly M3097 N1255)
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45
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0, 1, 3, 22, 333, 355, 103993, 104348, 208341, 312689, 833719, 1146408, 4272943, 5419351, 80143857, 165707065, 245850922, 411557987, 1068966896, 2549491779, 6167950454, 14885392687, 21053343141, 1783366216531, 3587785776203, 5371151992734, 8958937768937
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OFFSET
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0,3
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COMMENTS
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K. S. Lucas found, by brute-force search, using Maple programming, several different variants of integral identities which relate each of several first Pi convergents (A002485(n)/A002486(n)) to Pi.
I conjecture the following identity below, which represents a generalization of Stephen Lucas's experimentally obtained identities:
(-1)^n*(Pi-A002485(n)/A002486(n)) = (1/abs(i)*2^j)*Integral_{x=0..1} (x^l*(1-x)^m*(k+(k+i)*x^2)/(1+x^2)) dx where {i, j, k, l, m} are some integers (see the Mathematics Stack Exchange link below).
(End)
From a(1)=1 on also: Numbers for which |tan x| decreases monotonically to zero, in the same spirit asA004112,A046947,... -M. F. Hasler,Apr 01 2013
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REFERENCES
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P. Beckmann, A History of Pi. Golem Press, Boulder, CO, 2nd ed., 1971, p. 171 (but beware errors).
CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 88.
P. Finsler, Über die Faktorenzerlegung natuerlicher Zahlen, Elemente der Mathematik, 2 (1947), 1-11, see p. 7.
K. H. Rosen et al., eds., Handbook of Discrete and Combinatorial Mathematics, CRC Press, 2000; p. 293.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Eric Weisstein's World of Mathematics,Pi
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EXAMPLE
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The convergents are 0, 1, 3, 22/7, 333/106, 355/113, 103993/33102, 104348/33215, 208341/66317, 312689/99532, 833719/265381, 1146408/364913, 4272943/1360120, 5419351/1725033, 80143857/25510582, 165707065/52746197, 245850922/78256779, 411557987/131002976, 1068966896/340262731, 2549491779/811528438,... =A002485/A002486
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MAPLE
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Digits:= 60: E:= Pi; convert(evalf(E), confrac, 50, 'cvgts'): cvgts;
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MATHEMATICA
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PROG
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(PARI) contfracpnqn(cf=contfrac(Pi), #cf)[1, ] \\M. F. Hasler,Apr 01 2013, simplified Oct 13 2020
(PARI) e=9e9; for(n=1, 1e9, abs(tan(n))<e &&!print1(n "," ) && e=abs(tan(n))) \\ Illustration of |tan a(n)| -> 0 monotonically. -M. F. Hasler,Apr 01 2013
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CROSSREFS
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Cf.A096456(numerators of convergents to Pi/2).
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KEYWORD
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nonn,easy,nice,frac
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AUTHOR
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EXTENSIONS
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Extended and corrected by David Sloan, Sep 23 2002
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STATUS
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approved
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