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A157259 Decimal expansion of 7 - 2*sqrt(2).

%I A157259 #28 Oct 19 2024 15:57:32
%S A157259 4,1,7,1,5,7,2,8,7,5,2,5,3,8,0,9,9,0,2,3,9,6,6,2,2,5,5,1,5,8,0,6,0,3,
%T A157259 8,4,2,8,6,0,6,5,6,2,4,9,2,4,6,1,0,3,8,5,3,6,4,6,6,4,0,5,2,4,0,1,8,5,
%U A157259 3,5,0,4,3,0,7,5,7,8,5,9,2,2,2,9,9,2,2,4,9,3,1,3,4,4,7,1,6,8,5,4,5,2,9,9,7
%N A157259 Decimal expansion of 7 - 2*sqrt(2).
%C A157259 lim_{n -> infinity} b(n)/b(n-1) = (7+2*sqrt(2))/(7-2*sqrt(2)) for n mod 3 = {1, 2}, b = A129288.
%C A157259 lim_{n -> infinity} b(n)/b(n-1) = (7+2*sqrt(2))/(7-2*sqrt(2)) for n mod 3 = {0, 2}, b = A157257.
%H A157259 G. C. Greubel, <a href="/A157259/b157259.txt">Table of n, a(n) for n = 1..5000</a>
%H A157259 Pierre-Antoine Guihéneuf, <a href="http://images-archive.math.cnrs.fr/Rotations-discretes.html">Rotations Discrètes</a>, Images des Mathématiques, CNRS, 2018. See 3-2*sqrt(2), the fractional part of this constant, about the loss of information when rotating an image.
%H A157259 <a href="/index/Al#algebraic_02">Index entries for algebraic numbers, degree 2</a>
%F A157259 Equals 3 + Sum_{k>=0} binomial(2*k,k)/((k+1) * 8^k). - _Amiram Eldar_, Aug 03 2020
%F A157259 Equals 4 + exp(-arccosh(3)). - _Amiram Eldar_, Jul 06 2023
%F A157259 Equals 4 + (2-sqrt(2))/(2+sqrt(2)). - _Davide Rotondo_, Jun 08 2024
%e A157259 7 - 2*sqrt(2) = 4.17157287525380990239...
%t A157259 RealDigits[7-2*Sqrt[2],10,120][[1]] (* _Harvey P. Dale_, May 01 2012 *)
%o A157259 (PARI) 7 - 2*sqrt(2) \\ _G. C. Greubel_, Nov 28 2017
%o A157259 (Magma) [7 - 2*Sqrt(2)]; // _G. C. Greubel_, Nov 28 2017
%Y A157259 Cf. A129288, A157257, A157258 (decimal expansion of 7+2*sqrt(2)), A157260 (decimal expansion of (7+2*sqrt(2))/(7-2*sqrt(2))).
%K A157259 cons,nonn
%O A157259 1,1
%A A157259 _Klaus Brockhaus_, Feb 26 2009