A188722 - OEIS

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

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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]