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%I A016630 #34 Mar 21 2024 16:07:45
%S A016630 1,9,4,5,9,1,0,1,4,9,0,5,5,3,1,3,3,0,5,1,0,5,3,5,2,7,4,3,4,4,3,1,7,9,
%T A016630 7,2,9,6,3,7,0,8,4,7,2,9,5,8,1,8,6,1,1,8,8,4,5,9,3,9,0,1,4,9,9,3,7,5,
%U A016630 7,9,8,6,2,7,5,2,0,6,9,2,6,7,7,8,7,6,5,8,4,9,8,5,8,7,8,7,1,5,2
%N A016630 Decimal expansion of log(7).
%D A016630 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 2.
%D A016630 Uhler, Horace S.; Recalculation and extension of the modulus and of the logarithms of 2, 3, 5, 7 and 17. Proc. Nat. Acad. Sci. U. S. A. 26, (1940). 205-212.
%H A016630 Harry J. Smith, <a href="/A016630/b016630.txt">Table of n, a(n) for n = 1..20000</a>
%H A016630 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A016630 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F A016630 From _Peter Bala_, Nov 11 2019: (Start)
%F A016630 log(7) = 2*sqrt(3)*Integral_{t = 0..sqrt(3)/3} (1 - t^4)/(1 + t^6) dt.
%F A016630 log(7) = (8/9)*Sum_{n >= 0} (12*n+11)/((6*n+1)*(6*n+5))*(-1/27)^n.
%F A016630 log(7) = 6*Sum_{n >= 0} ( 243/(12*n+1) - 27/(12*n+5) - 9/(12*n+7) + 1/(12*n+11) )*(1/729)^(n+1), a BPP-type formula. (End)
%F A016630 log(7) = 2*Sum_{n >= 1} 1/(n*P(n, 4/3)*P(n-1, 4/3)), where P(n, x) denotes the n-th Legendre polynomial. The first 20 terms of the series gives the approximation log(7) = 1.945910149055(27...), correct to 12 decimal places. - _Peter Bala_, Mar 18 2024
%e A016630 1.945910149055313305105352743443179729637084729581861188459390149937579...
%t A016630 First[RealDigits[Log[7], 10, 100]] (* _Paolo Xausa_, Mar 21 2024 *)
%o A016630 (PARI) default(realprecision, 20080); x=log(7); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b016630.txt", n, " ", d)); \\ _Harry J. Smith_, May 16 2009
%Y A016630 Cf. A016735 Continued fraction.
%K A016630 nonn,cons
%O A016630 1,2
%A A016630 _N. J. A. Sloane_

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