A188883 Decimal expansion of (1 + sqrt(1 + Pi^2))/Pi.

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

Offset

1,2

Comments

Decimal expansion of the length/width ratio of a (2/Pi)-extension rectangle. See A188640 for definitions of shape and r-extension rectangle.

A (2/Pi)-extension rectangle matches the continued fraction [1,2,1,2,1,1,3,1,1,5,1,7,1,1,23,2,...] for the shape L/W = (1 + sqrt(1 + Pi^2))/Pi. This is analogous to the matching of a golden rectangle to the continued fraction [1,1,1,1,1,1,1,1,...]. Specifically, for the (2/Pi)-extension rectangle, 1 square is removed first, then 2 squares, then 1 square, then 2 squares, ..., so that the original rectangle of shape (1 + sqrt(1 + Pi^2))/Pi is partitioned into an infinite collection of squares.

Links

Table of n, a(n) for n=1..130.

Index entries for transcendental numbers

Example

1.36774839493136744469969176568220545565111326890...

Mathematica

r = 2/Pi; t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]
ContinuedFraction[t, 120]

Prog

(PARI) (sqrt(Pi^2+1)+1)/Pi \\ Charles R Greathouse IV, Oct 01 2022

Crossrefs

Cf. A188640, A188884, A188724, A188726.

Sequence in context: A290943 A067753 A129023 * A336086 A152083 A251532

Adjacent sequences: A188880 A188881 A188882 * A188884 A188885 A188886

Keyword

nonn,cons

Author

Clark Kimberling, Apr 12 2011

Status

approved