A164833 Decimal expansion of Pi/8 - log(2)/2.
0, 4, 6, 1, 2, 5, 4, 9, 1, 4, 1, 8, 7, 5, 1, 5, 0, 0, 0, 9, 9, 2, 1, 4, 3, 6, 2, 1, 8, 0, 8, 4, 9, 5, 7, 6, 4, 8, 6, 8, 9, 6, 1, 0, 7, 7, 4, 1, 7, 6, 0, 6, 0, 0, 5, 6, 1, 5, 2, 8, 0, 6, 9, 2, 9, 1, 7, 8, 0, 2, 3, 9, 8, 0, 0, 9, 2, 8, 7, 6, 7, 0, 2, 5, 5, 7, 2, 6, 8, 9, 6, 6, 9, 5, 5, 5, 2, 8, 9, 7, 2, 6, 7, 6, 7, 7, 7, 0, 3, 0, 3, 8, 7, 4, 9, 4, 5, 4, 6
Offset: 0
Examples
0.0461254914187515000992143621808495764868961077417606... 1/(2*3*4) + 1/(6*7*8) + 1/(10*11*12) + 1/(14*15*16) + ... [_Bruno Berselli_, Mar 17 2014]
References
- Mohammad K. Azarian, Problem 1218, Pi Mu Epsilon Journal, Vol. 13, No. 2, Spring 2010, p. 116. Solution published in Vol. 13, No. 3, Fall 2010, pp. 183-185.
- L. B. W. Jolley, Summation of series, Dover Publications Inc. (New York), 1961, p. 46 (series n. 251).
- A. J. Van Der Poorten, Effectively computable bounds for the solutions of certain Diophantine equations, Acta Arith., 33 (1977), pp. 195-207.
Links
- G. C. Greubel, Table of n, a(n) for n = 0..10000
- Michel Waldschmidt, Perfect Powers: Pillai's works and their developments, Aug 27, 2009.
Crossrefs
Programs
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Magma
SetDefaultRealField(RealField(130)); R:= RealField(); (Pi(R)-4*Log(2))/8; // G. C. Greubel, Aug 11 2019
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Maple
evalf[130]((Pi - 4*log(2))/8 ); # G. C. Greubel, Aug 11 2019
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Mathematica
Join[{0},RealDigits[Pi/8-Log[2]/2,10,120][[1]]] (* Harvey P. Dale, Nov 13 2012 *) -
PARI
default(realprecision, 130); (Pi - 4*log(2))/8 \\ G. C. Greubel, Aug 11 2019
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Sage
numerical_approx((pi-4*log(2))/8, digits=130) # G. C. Greubel, Aug 11 2019
Formula
Equals Sum_{n>=0} Sum_{m>=0} 1/((4*n+3)^(2*m+1)).
Equals Sum_{k>=1} 1/( (4*k-2)*(4*k-1)*(4*k) ). - Bruno Berselli, Mar 17 2014
Extensions
Normalized offset and leading zeros - R. J. Mathar, Sep 27 2009
Comments