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A156164 Decimal expansion of 17 + 12*sqrt(2).

%I A156164 #33 Aug 21 2023 10:15:13
%S A156164 3,3,9,7,0,5,6,2,7,4,8,4,7,7,1,4,0,5,8,5,6,2,0,2,6,4,6,9,0,5,1,6,3,7,
%T A156164 6,9,4,2,8,3,6,0,6,2,5,0,4,5,2,3,3,7,6,8,7,8,1,2,0,1,5,6,8,5,5,8,8,8,
%U A156164 7,8,9,7,4,1,5,4,5,2,8,4,4,6,6,2,0,4,6,5,0,4,1,1,9,3,1,6,9,8,8,7,2,8,2,0,1
%N A156164 Decimal expansion of 17 + 12*sqrt(2).
%C A156164 Lim_{n -> infinity} b(n)/b(n-1) = 17+12*sqrt(2) for b = A156160, A156161, A156162, A278310.
%C A156164 Conjecturally, the fractional part 0.97056 27484 ... of this constant equals ( (1 + 2 * Sum_{n >= 1} (-1)^n*exp(-2*Pi*n^2))/(1 + 2 * Sum_{n >= 1} exp(-2*Pi*n^2)) )^4. The series are rapidly converging. For example, summing both series from n = 1 to n = 2 approximates the fractional part of the constant as ( (1 - 2*exp(-2*Pi) + 2*exp(-8*Pi))/(1 + 2*exp(-2*Pi) + 2*exp(-8*Pi)) )^4 = 0.97056 27484 77140 58562 026(89) ..., correct to 23 decimal places. - _Peter Bala_, Jun 05 2019
%H A156164 G. C. Greubel, <a href="/A156164/b156164.txt">Table of n, a(n) for n = 2..10000</a>
%H A156164 M. Kontsevich and D. Zagier, <a href="http://www.ihes.fr/~maxim/TEXTS/Periods.pdf">Periods</a>, Institut des Hautes Etudes Scientifiques 2001 IHES/M/01/22. Published in B. Engquist and W. Schmid, editors, Mathematics Unlimited - 2001 and Beyond, 2 vols., Springer-Verlag, 2001, pp. 771-808, section 2.4. Example 4.
%H A156164 <a href="/index/Al#algebraic_02">Index entries for algebraic numbers, degree 2</a>
%F A156164 17+12*sqrt(2) = (3+2*sqrt(2))^2 = (1+sqrt(2))^4. - _Klaus Brockhaus_, Feb 14 2009. (corrected by _Bruno Berselli_, Feb 19 2013)
%e A156164 17 + 12*sqrt(2) = 33.97056274847714058562026469051637694283606250452337687...
%t A156164 RealDigits[17 + 12 Sqrt[2], 10, 120][[1]] (* _Harvey P. Dale_, Nov 30 2014 *)
%o A156164 (PARI) 17+sqrt(288) \\ _Charles R Greathouse IV_, May 07 2015
%Y A156164 Cf. A002193: decimal expansion of sqrt(2); A156035: decimal expansion of 3+2*sqrt(2); A156163: decimal expansion of (19+6*sqrt(2))/17.
%Y A156164 Cf. A005259, A059415/A059416.
%K A156164 nonn,cons
%O A156164 2,1
%A A156164 _Klaus Brockhaus_, Feb 09 2009