A188734 -id:A188734 - cp's OEIS frontend

A188941 Decimal expansion of (9+sqrt(65))/4.

4, 2, 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 14 2011

Apart from the first digit, the same as A171417. Apart from the first 2 digits, the same as A188734. - R. J. Mathar, Apr 15 2011
Decimal expansion of the shape (= length/width = (9+sqrt(65))/4) of the greater (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.

			4.2655644370746374130916533075759427827835990...

Cf. A188738, A188940.

r = 9/2; t = (r + (-4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]
RealDigits[(9+Sqrt[65])/4,10,150][[1]] (* Harvey P. Dale, Jan 31 2023 *)

(9+sqrt(65))/4 \\ Jinyuan Wang, Apr 14 2020

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

Original entry on oeis.org

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

Clark Kimberling, Apr 12 2011

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.

			4.712214450449026180436552853729406120424034071860691042930...

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

Cf. A188640, A188734.

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

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

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

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

cp's OEIS Frontend

A188941 Decimal expansion of (9+sqrt(65))/4.

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