A188640 -id:A188640 - cp's OEIS frontend

A188732 Decimal expansion of (5+sqrt(61))/6.

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

Offset: 1

Clark Kimberling, Apr 10 2011

Decimal expansion of shape of a (5/3)-extension rectangle; see A188640 for definitions of shape and r-extension rectangle. Briefly, shape=length/width, and an r-extension rectangle is composed of two rectangles of shape r.

The continued fractions of the constant are 2, 7, 2, 2, 7, 2, 2, 7, 2, 2, 7, 2, 2,...

			2.1350416126511090656882871226265169022613841894414272166962...

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

Cf. A188640.

SetDefaultRealField(RealField(100)); (5+Sqrt(61))/6; // G. C. Greubel, Nov 01 2018

evalf((5+sqrt(61))/6,140); # Muniru A Asiru, Nov 01 2018

RealDigits[(5 + Sqrt[61])/6, 10, 111][[1]] (* Robert G. Wilson v, Aug 18 2011 *)

(sqrt(61)+5)/16 \\ Charles R Greathouse IV, Apr 25 2016

A188733 Decimal expansion of (9+sqrt(145))/8.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 10 2011

Decimal expansion of shape of a (9/4)-extension rectangle; see A188640 for definitions of shape and r-extension rectangle. Briefly, shape=length/width, and an r-extension rectangle is composed of two rectangles of shape r. The (periodic) continued fraction of the constant is [2,1,1,1,2,2,1,1,1,2,2,1,...].

			2.6301993223490369350160301287973260065531690506317...

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

Cf. A188640.

SetDefaultRealField(RealField(100)); (9+Sqrt(145))/8; // G. C. Greubel, Nov 01 2018

evalf((9+sqrt(145))/8,120); # Muniru A Asiru, Nov 01 2018

r = 9/4; t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]
ContinuedFraction[t, 120]
RealDigits[(9+Sqrt[145])/8,10,120][[1]] (* Harvey P. Dale, Jul 20 2025 *)

(9+sqrt(145))/8 \\ Michel Marcus, Sep 03 2014

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

A188736 Decimal expansion of (3+sqrt(34))/5.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 12 2011

Decimal expansion of the length/width ratio of a (6/5)-extension rectangle. See A188640 for definitions of shape and r-extension rectangle.

A (6/5)-extension rectangle matches the continued fraction [1,1,3,3,1,1,1,1,3,3,1,1,1,1,3,3,...] for the shape L/W=(3+sqrt(34))/5. 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 (6/5)-extension rectangle, 1 square is removed first, then 1 square, then 3 squares, then 3 squares,..., so that the original rectangle of shape (3+sqrt(34))/5 is partitioned into an infinite collection of squares.

			1.76619037896906009417483057550911661530...

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

Cf. A188640.

SetDefaultRealField(RealField(100)); (3 + Sqrt(34))/5; // G. C. Greubel, Nov 01 2018

evalf((3+sqrt(34))/5,140); # Muniru A Asiru, Nov 01 2018

RealDigits[(3 + Sqrt[34])/5, 10, 111][[1]] (* Robert G. Wilson v, Aug 18 2011 *)

(sqrt(34)+3)/5 \\ Charles R Greathouse IV, Apr 25 2016

A188796 Continued fraction of e+sqrt(1+e^2).

Original entry on oeis.org

5, 1, 1, 1, 1, 2, 7, 1, 7, 3, 1, 5, 2, 5, 1, 1, 1, 3, 6, 8, 26, 2, 1, 2, 3, 1, 1, 1, 13, 1, 10, 2, 5, 1, 10, 1, 1, 4, 1, 1, 2, 1, 3, 3, 2, 7, 1, 2, 21, 1, 1, 1, 1, 3, 2, 8, 9, 4, 2, 8, 1, 2, 1, 7, 1, 1, 19, 1, 4, 9, 1, 2, 1, 4, 2, 1, 4, 1, 4, 6, 2, 5, 10, 1, 2, 2, 10, 1, 1, 25, 1, 4, 13, 9, 2, 1, 2, 4, 8, 1, 1, 33, 1, 2, 1, 1, 1, 21, 1, 3, 1, 18, 1, 6, 43, 2, 1, 1, 1, 8

Offset: 0

Clark Kimberling, Apr 10 2011

See A188640 for the origin of this constant.

Cf. A188640 (decimal expansion), A188721.

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

e+sqrt(1+e^2) = 5.614668.. = [5,1,1,1,1,2,7,1,7,3,1,5,2,5,1,1,1,...].

Offset changed by Andrew Howroyd, Jul 07 2024

A188883 Decimal expansion of (1 + sqrt(1 + Pi^2))/Pi.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 12 2011

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

A (2/Pi)-extension rectangle matches the continued fraction [1,2,1,2,1,1,3,1,1,5,1,7,1,1,23,2,...] for the shape L/W = (1 + sqrt(1 + Pi^2))/Pi. 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 (2/Pi)-extension rectangle, 1 square is removed first, then 2 squares, then 1 square, then 2 squares, ..., so that the original rectangle of shape (1 + sqrt(1 + Pi^2))/Pi is partitioned into an infinite collection of squares.

			1.36774839493136744469969176568220545565111326890...

Index entries for transcendental numbers

Cf. A188640, A188884, A188724, A188726.

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

(sqrt(Pi^2+1)+1)/Pi \\ Charles R Greathouse IV, Oct 01 2022

A188884 Continued fraction of (1 + sqrt(1 + Pi^2))/Pi.

Original entry on oeis.org

1, 2, 1, 2, 1, 1, 3, 1, 1, 5, 1, 7, 1, 1, 23, 2, 2, 4, 3, 1, 11, 158, 1, 1, 1, 1, 4, 2, 1, 6, 2, 19, 75, 1, 1, 1, 28, 1, 29, 6, 8, 1, 5, 1, 4, 2, 1, 8, 1, 1, 19, 1, 1, 9, 2, 2, 3, 1, 2, 11, 1, 1, 3, 1, 1, 4, 169, 1, 1, 2, 1, 3, 1, 1, 10, 2, 1, 3, 8, 2, 4, 8, 5, 1, 8, 1, 7, 1, 1, 1, 1, 4, 38, 1, 5, 1, 43, 1, 1, 1, 1, 2, 1, 8, 1, 20, 1, 1, 1, 2, 13, 51, 2, 21, 1, 2, 5, 1, 1, 1

Offset: 0

Clark Kimberling, Apr 12 2011

For a geometric interpretation, see A188640 and A188883.

			(1 + sqrt(1 + Pi^2))/Pi = [1, 2, 1, 2, 1, 1, 3, 1, 1, 5, 1, 7, 1, 1, 23, 2, ...].

Cf. A188640, A188883 (decimal expansion), A188726.

r = 2/Pi; t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]
ContinuedFraction[t, 120]
ContinuedFraction[(1+Sqrt[1+Pi^2])/Pi,120] (* Harvey P. Dale, May 02 2025 *)

Offset changed by Andrew Howroyd, Jul 07 2024

A188885 Decimal expansion of (1+sqrt(1+e^2))/e.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 12 2011

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

A (2/e)-extension rectangle matches the continued fraction [1,2,3,3,1,15,1,3,7,1,2,4,...] for the shape L/W=(1+sqrt(1+e^2))/e. 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 (2/e)-extension rectangle, 1 square is removed first, then 2 squares, then 3 square2, then 3 squares,..., so that the original rectangle of shape (1+sqrt(1+e^2))/e is partitioned into an infinite collection of squares.

			1.433400573405154846331906919976752328843391...

Cf. A188640, A188886.

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

A188886 Continued fraction of (1+sqrt(1+e^2))/e.

Original entry on oeis.org

1, 2, 3, 3, 1, 15, 1, 3, 7, 1, 2, 4, 2, 1, 4, 1, 1, 12, 4, 52, 1, 2, 1, 7, 3, 6, 1, 21, 1, 11, 1, 4, 1, 3, 2, 3, 2, 7, 1, 1, 2, 1, 3, 2, 1, 43, 4, 1, 1, 4, 4, 18, 2, 4, 4, 2, 1, 2, 3, 1, 3, 9, 1, 9, 4, 1, 6, 1, 1, 1, 2, 10, 1, 1, 1, 1, 2, 1, 2, 1, 2, 21, 2, 2, 1, 4, 1, 3, 12, 1, 9, 6, 1, 1, 4, 5, 2, 1, 1, 1, 1, 3, 1, 3, 16, 1, 6, 3, 10, 1, 8, 1, 8, 1, 13, 21, 1, 2, 4, 1

Offset: 0

Clark Kimberling, Apr 12 2011

For a geometric interpretation, see A188640 and A188885.

			(1+sqrt(1+e^2))/e=[1,2,3,3,1,15,1,3,7,1,2,4,...].

Cf. A188640, A188885 (decimal expansion).

r = 2/E; t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]
ContinuedFraction[t, 120]
ContinuedFraction[(1+Sqrt[1+E^2])/E,120] (* Harvey P. Dale, Feb 02 2025 *)

Offset changed by Andrew Howroyd, Jul 08 2024

A188922 Decimal expansion of (sqrt(3) + sqrt(7))/2.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 13 2011

Decimal expansion of the length/width ratio of a sqrt(3)-extension rectangle. See A188640 for definitions of shape and r-extension rectangle.

A sqrt(3)-extension rectangle matches the continued fraction [2,5,3,2,2,9,1,2,1,2,1,9,...] for the shape L/W=(sqrt(3)+sqrt(7))/2. 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 sqrt(3)-extension rectangle, 2 squares are removed first, then 5 squares, then 3 squares, then 2 squares, ..., so that the original rectangle of shape (sqrt(3)+sqrt(7))/2 is partitioned into an infinite collection of squares.

			2.1889010593167339420145310475725663963265322184...

Cf. A188640, A188923.

r = 3^(1/2); t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]
ContinuedFraction[t, 120]
RealDigits[(Sqrt[3]+Sqrt[7])/2,10,140][[1]] (* Harvey P. Dale, Feb 27 2023 *)

(sqrt(3)+sqrt(7))/2 = exp(asinh(cos(Pi/6))). - Geoffrey Caveney, Apr 23 2014

cos(Pi/6) + sqrt(1+cos(Pi/6)^2). - Geoffrey Caveney, Apr 23 2014

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A188732 Decimal expansion of (5+sqrt(61))/6.

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A188733 Decimal expansion of (9+sqrt(145))/8.

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

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A188736 Decimal expansion of (3+sqrt(34))/5.

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A188796 Continued fraction of e+sqrt(1+e^2).

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A188883 Decimal expansion of (1 + sqrt(1 + Pi^2))/Pi.

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A188884 Continued fraction of (1 + sqrt(1 + Pi^2))/Pi.

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