A188722 Decimal expansion of (Pi+sqrt(4+Pi^2))/2.

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

Offset

1,1

Comments

Decimal expansion of shape of a Pi-extension rectangle; see A188640 for definitions of shape and r-extension rectangle. Briefly, an r-extension rectangle is composed of two rectangles having shape r.

A Pi-extension rectangle matches the continued fraction A188723 of the shape L/W = (Pi+sqrt(4+Pi^2))/2. This is analogous to the matching of a golden rectangle to the continued fraction [1,1,1,1,1,1,1,...]. Specifically, for a Pi-extension rectangle, 3 squares are removed first, then 2 squares, then 3 squares, then 4 squares, then 2 squares,..., so that the original rectangle is partitioned into an infinite collection of squares.

Links

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

Index entries for transcendental numbers

Formula

(Pi+sqrt(4+Pi^2))/2 = [Pi,Pi,Pi,...] (continued fraction). - Clark Kimberling, Sep 23 2013

Example

3.4328922159134832442014603702358109669027341058202444195...

Mathematica

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

Prog

(PARI) (Pi+sqrt(4+Pi^2))/2 \\ Michel Marcus, Apr 01 2015

Crossrefs

Cf. A188640, A188723, A188720, A000796.

Sequence in context: A281975 A133617 A199286 * A257526 A038774 A092910

Adjacent sequences: A188719 A188720 A188721 * A188723 A188724 A188725

Keyword

nonn,cons

Author

Clark Kimberling, Apr 09 2011

Status

approved