A188724 Decimal expansion of shape of a (Pi/2)-extension rectangle; shape = (1/4)*(Pi + sqrt(16 + Pi^2)).

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

Offset: 1

Author

Clark Kimberling, Apr 09 2011

Keywords

• nonn
• easy
• cons

Comments

See A188640 for definitions of shape and r-extension rectangle. Briefly, an r-extension rectangle is composed of two rectangles of shape r.

A (Pi/2)-extension rectangle matches the continued fraction [2,17,1,1,3,1,3,2,2,1637,1,210,7,...] of the shape L/W = (1/4)*(Pi + sqrt(16 + Pi^2)). This is analogous to the matching of a golden rectangle to the continued fraction [1,1,1,1,1,1,1,...]. Specifically, for the (Pi/2)-extension rectangle, 2 squares are removed first, then 17 squares, then 1 square, then 1 square, then 3 squares, ..., so that the original rectangle is partitioned into an infinite collection of squares.

Examples

			2.0569524387109659093967879243788072585880991...
		

Links

• Index entries for transcendental numbers

Crossrefs

Cf. A188640, A188722.

Programs

Mathematica

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

Extensions

a(130) corrected by Georg Fischer, Jul 16 2021