login
A320639
Decimal expansion of (C + sqrt(4 + C^2))/2, where C is the Catalan constant.
2
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, 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, 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
OFFSET
1,2
COMMENTS
Decimal expansion of the shape of a Catalan-extension rectangle; see A188640 for definitions of shape and r-extension rectangle.
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.
LINKS
Clark Kimberling, Two kinds of golden triangles, generalized to match continued fractions, Journal for Geometry and Graphics, 11 (2007) 165-171.
EXAMPLE
1.557868355876025567309823249177... = [Catalan, Catalan, Catalan, ...]
MAPLE
evalf((Catalan+sqrt(4+Catalan^2))/2, 135);
MATHEMATICA
First@ RealDigits[(Catalan + Sqrt[4 + Catalan^2])/2, 10, 105] (* Michael De Vlieger, Oct 23 2018 *)
PROG
(PARI) (Catalan+sqrt(4+Catalan^2))/2 \\ Felix Fröhlich, Oct 23 2018
(Magma) R:= RealField(200); (Catalan(R) + Sqrt(4 + Catalan(R)^2)) / 2; // Vincenzo Librandi, Oct 24 2018
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Paolo P. Lava, Oct 18 2018
STATUS
approved