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Decimal expansion of (C + sqrt(4 + C^2))/2, where C is the Catalan constant.
2

%I #26 Sep 08 2022 08:46:23

%S 1,5,5,7,8,6,8,3,5,5,8,7,6,0,2,5,5,6,7,3,0,9,8,2,3,2,4,9,1,7,7,4,0,6,

%T 9,9,0,6,9,7,1,6,4,3,1,0,8,6,0,1,3,3,6,0,2,3,2,1,4,7,9,8,0,1,4,0,5,9,

%U 5,6,7,1,1,2,7,4,4,7,4,0,4,8,3,1,9,9,0,7,7,2,5,6,6,2,0,9,2,9,4,1,5,5,9,4,5,5,2,9,9,1,3,3,3,3,9,2,3,4,3,3,7,0,4,6,6,9,5,9,0,9,4

%N Decimal expansion of (C + sqrt(4 + C^2))/2, where C is the Catalan constant.

%C Decimal expansion of the shape of a Catalan-extension rectangle; see A188640 for definitions of shape and r-extension rectangle.

%C Specifically, for a Catalan-extension rectangle, 1 square is removed first, then 1 square, then 1 square again, then 3 squares, then 1 square, ... (see A320640), so that the original rectangle is partitioned into an infinite collection of squares.

%H G. C. Greubel, <a href="/A320639/b320639.txt">Table of n, a(n) for n = 1..10000</a>

%H Clark Kimberling, <a href="http://www.heldermann.de/JGG/JGG11/JGG112/jgg11014.htm">Two kinds of golden triangles, generalized to match continued fractions</a>, Journal for Geometry and Graphics, 11 (2007) 165-171.

%e 1.557868355876025567309823249177... = [Catalan, Catalan, Catalan, ...]

%p evalf((Catalan+sqrt(4+Catalan^2))/2,135);

%t First@ RealDigits[(Catalan + Sqrt[4 + Catalan^2])/2, 10, 105] (* _Michael De Vlieger_, Oct 23 2018 *)

%o (PARI) (Catalan+sqrt(4+Catalan^2))/2 \\ _Felix Fröhlich_, Oct 23 2018

%o (Magma) R:= RealField(200); (Catalan(R) + Sqrt(4 + Catalan(R)^2)) / 2; // _Vincenzo Librandi_, Oct 24 2018

%Y Cf. A006752, A188640, A320640.

%K nonn,cons

%O 1,2

%A _Paolo P. Lava_, Oct 18 2018