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A224578 Decimal expansion of (gamma+sqrt(4+gamma^2))/2, where gamma is the Euler-Mascheroni constant.

%I A224578 #19 Sep 08 2022 08:46:04
%S A224578 1,3,2,9,4,2,2,1,6,7,9,3,6,1,7,3,5,8,1,8,7,9,4,1,7,7,6,8,1,0,5,6,3,6,
%T A224578 2,4,4,8,0,8,4,9,5,8,3,3,2,9,2,0,0,0,8,3,0,4,4,2,6,2,1,4,6,5,7,4,2,5,
%U A224578 8,1,9,9,6,9,1,3,2,6,1,7,8,1,2,2,7,6,7
%N A224578 Decimal expansion of (gamma+sqrt(4+gamma^2))/2, where gamma is the Euler-Mascheroni constant.
%C A224578 Decimal expansion of shape of a gamma-extension rectangle; see A188640 for definitions of shape and r-extension rectangle.
%C A224578 Specifically, for a gamma-extension rectangle, 1 square is removed first, then 3 squares, then 28 squares, then 13 squares, then 3 squares,...(see A224579), so that the original rectangle is partitioned into an infinite collection of squares.
%H A224578 Paolo P. Lava, <a href="/A224578/b224578.txt">Table of n, a(n) for n = 1..200</a>
%H A224578 Clark Kimberling, <a href="http://www.ams.org/mathscinet-getitem?mr=2420629">Two kinds of golden triangles, generalized to match continued fractions</a>, Journal for Geometry and Graphics, 11 (2007) 165-171
%e A224578 1.329422167936173581879417768105... = [gamma, gamma, gamma, ...]
%p A224578 evalf((gamma+sqrt(4+gamma^2))/2,90);
%t A224578 RealDigits[(EulerGamma + Sqrt[4 + EulerGamma^2])/2, 10, 100][[1]] (* _G. C. Greubel_, Aug 30 2018 *)
%o A224578 (PARI) Euler/2+sqrt(4+Euler^2)/2 \\ _Charles R Greathouse IV_, Dec 11 2013
%o A224578 (Magma) SetDefaultRealField(RealField(100)); R:= RealField(); (EulerGamma(R) + Sqrt(4 + EulerGamma(R)^2))/2; // _G. C. Greubel_, Aug 30 2018
%Y A224578 Cf. A001620, A188640, A224579.
%K A224578 nonn,cons
%O A224578 1,2
%A A224578 _Paolo P. Lava_, Apr 11 2013