A188941 -id:A188941 - cp's OEIS frontend

A188734 Decimal expansion of (7+sqrt(65))/4.

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

Offset: 1

Clark Kimberling, Apr 12 2011

Apart from the second digit, the same as A171417. - R. J. Mathar, Apr 15 2011
Apart from the first two digits, the same as A188941. - Joerg Arndt, Apr 16 2011
Decimal expansion of the length/width ratio of a (7/2)-extension rectangle.  See A188640 for definitions of shape and r-extension rectangle.
A (7/2)-extension rectangle matches the continued fraction [3,1,3,3,1,3,3,1,3,3,1,3,3,...] for the shape L/W=(7+sqrt(65))/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 (7/2)-extension rectangle, 3 squares are removed first, then 1 square, then 3 squares, then 3 squares,..., so that the original rectangle of shape (7+sqrt(65))/4 is partitioned into an infinite collection of squares.

			3.7655644370746374130916533075759427827835990...

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

Cf. A188640.

SetDefaultRealField(RealField(100)); (7+Sqrt(65))/4; // G. C. Greubel, Nov 01 2018

evalf((7+sqrt(65))/4,140); # Muniru A Asiru, Nov 01 2018

r = 7/2; t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]

default(realprecision, 100); (7+sqrt(65))/4 \\ G. C. Greubel, Nov 01 2018

A188940 Decimal expansion of (9-sqrt(65))/4.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 14 2011

Decimal expansion of the shape (= length/width = (9-sqrt(65))/4) of the lesser (9/2)-contraction rectangle.

See A188738 for an introduction to lesser and greater r-contraction rectangles, their shapes, and partitioning these rectangles into a sets of squares in a manner that matches the continued fractions of their shapes.

			0.23443556292536258690834669242405721721640...

Cf. A188941, A188738.

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

cp's OEIS Frontend

A188734 Decimal expansion of (7+sqrt(65))/4.

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