A188723 -id:A188723 - cp's OEIS frontend

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

Clark Kimberling, Apr 09 2011

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.

			3.4328922159134832442014603702358109669027341058202444195...

Index entries for transcendental numbers

Cf. A188640, A188723, A188720, A000796.

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

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

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

cp's OEIS Frontend

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

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