A188735 Decimal expansion of (9+sqrt(97))/4.

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

Offset: 1

Author

Clark Kimberling, Apr 12 2011

Keywords

• nonn
• cons

Comments

Decimal expansion of the length/width ratio of a (9/2)-extension rectangle. See A188640 for definitions of shape and r-extension rectangle.

A (9/2)-extension rectangle matches the continued fraction [4,1,2,2,9,2,2,1,4,4,1,2,2,9,...] for the shape L/W=(9+sqrt(97))/4. 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 (9/2)-extension rectangle, 4 squares are removed first, then 1 square, then 2 squares, then 2 squares,..., so that the original rectangle of shape (9+sqrt(97))/4 is partitioned into an infinite collection of squares.

Examples

			4.712214450449026180436552853729406120424034071860691042930...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A188640, A188734.

Programs

Magma

SetDefaultRealField(RealField(100)); (9+Sqrt(97))/4; // G. C. Greubel, Nov 01 2018

Maple

evalf((9+sqrt(97))/4,140); # Muniru A Asiru, Nov 01 2018

Mathematica

r = 9/2; t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]
ContinuedFraction[t, 120]

PARI

(sqrt(97)+9)/4 \\ Charles R Greathouse IV, Apr 25 2016