A214726 -id:A214726 - cp's OEIS frontend

A188887 Decimal expansion of sqrt(2 + sqrt(3)).

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

Offset: 1

Clark Kimberling, Apr 12 2011

Decimal expansion of the length/width ratio of a sqrt(2)-extension rectangle.  See A188640 for definitions of shape and r-extension rectangle.
A sqrt(2)-extension rectangle matches the continued fraction [1,1,13,1,2,15,10,1,18,1,1,21,,...] (A188888) for the shape L/W = sqrt(2 + sqrt(3)). 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(2)-extension rectangle, 1 square is removed first, then 1 square, then 13 squares, then 1 square, ..., so that the original rectangle of shape sqrt(2 + sqrt(3)) is partitioned into an infinite collection of squares.
sqrt(2 + sqrt(3)) is also the shape of the greater sqrt(6)-contraction rectangle; see A188738.
This constant is also the length of the Steiner span of three vertices of a unit square. - Jean-François Alcover, May 22 2014
It is also the larger positive coordinate of (symmetrical) intersection points created by x^2 + y^2 = 4 circle and y = 1/x hyperbola. The smaller coordinate is A101263. - Leszek Lezniak, Sep 18 2018
Length of the shortest diagonal in a regular 12-gon with unit side. - Mohammed Yaseen, Nov 12 2020

			1.931851652578136573499486399457794735267809678016809...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Burkard Polster, Irrational roots, Mathologer video (2018).
Index entries for algebraic numbers, degree 4.

Cf. A019884, A019973, A091998, A101263, A188640, A188888, A214726, A217870, A329247.

Sqrt(2 + Sqrt(3)); // G. C. Greubel, Apr 10 2018

r = 2^(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[2 + Sqrt[3]], 10, 100][[1]] (* G. C. Greubel, Apr 10 2018 *)

sqrt(2 + sqrt(3)) \\ G. C. Greubel, Apr 10 2018

Equals (sqrt(6) + sqrt(2))/2.
Equals exp(asinh(cos(Pi/4))). - Geoffrey Caveney, Apr 23 2014
Equals cos(Pi/4) + sqrt(1 + cos(Pi/4)^2). - Geoffrey Caveney, Apr 23 2014
Equals i^(1/6) + i^(-1/6). - Gary W. Adamson, Jul 07 2022
Equals the largest root of x - 1/x = sqrt(2) and of x^2 + 1/x^2 = 4. - Gary W. Adamson, Jun 12 2023
Equals Product_{k>=0} ((12*k + 2)*(12*k + 10))/((12*k + 1)*(12*k + 11)). - Antonio Graciá Llorente, Feb 24 2024
From Amiram Eldar, Nov 23 2024: (Start)
Equals A214726 / 2 = 2 * A019884 = 1 / A101263 = exp(A329247) = A217870^2 = sqrt(A019973).
Equals Product_{k>=1} (1 - (-1)^k/A091998(k)). (End)

A246726 Decimal expansion of r_4, the 4th smallest radius < 1 for which a compact packing of the plane exists, with disks of radius 1 and r_4.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Sep 02 2014

			0.3491981862085498764736232370456943152782620498437475...

Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020-2021, p. 73.
Index entries for algebraic numbers, degree 4.

Cf. A246723 (r_1), A246724 (r_2), A246725 (r_3), A246727 (r_5), A002193 (r_6 = sqrt(2)-1), A246728 (r_7), A246729 (r_8), A246730 (r_9).

Cf. A214726.

RealDigits[Root[x^4 - 28x^3 - 10x^2 + 4x + 1, x, 3], 10, 103] // First

3rd root of x^4 - 28x^3 - 10x^2 + 4x + 1.

Equals 1/(cosec(Pi/12)-1) = 1/(A214726 - 1). - Amiram Eldar, Mar 27 2022

A337402 Decimal expansion of the length of third shortest diagonal in a regular 12-gon with unit edge length.

Original entry on oeis.org

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

Offset: 1

Mohammed Yaseen, Aug 26 2020

The distinct diagonal lengths in a regular 12-gon ABC...JKL with unit edge length are
AC = sqrt(2 +   sqrt(3)) =     sqrt(2)/(-1+sqrt(3)) = A188887
AD = sqrt(4 + 2*sqrt(3)) =          2 /(-1+sqrt(3)) = A090388
AE = sqrt(6 + 3*sqrt(3)) =     sqrt(6)/(-1+sqrt(3))
AF = sqrt(7 + 4*sqrt(3)) = (1+sqrt(3))/(-1+sqrt(3)) = A019973
AG = sqrt(8 + 4*sqrt(3)) =   2*sqrt(2)/(-1+sqrt(3)) = A214726

			3.34606521495123162230117512366749281383748155339375...

Paolo Xausa, Table of n, a(n) for n = 1..10000
N. J. A. Sloane and Gavin A. Theobald, On Dissecting Polygons into Rectangles, arXiv:2309.14866 [math.CO], 2023. See Eq. (2.3).
I. J. Zucker, G. S. Joyce, Special values of the hypergeometric series II, Math. Proc. Cam. Phil. Soc. 131 (2001) 309 eq (8.9)

Cf. A337301, A188887, A090388, A019973, A214726, A145439.

First[RealDigits[Sqrt[6+3Sqrt[3]],10,100]] (* Paolo Xausa, Oct 19 2023 *)

sqrt(6 + 3*sqrt(3)) \\ Michel Marcus, Aug 26 2020

Equals sin(Pi/3)/sin(Pi/12) = sqrt(2) + 2*cos(Pi/12) = sqrt(3*cot(Pi/12)).
Equals sqrt(6 + 3*sqrt(3)) = sqrt(6)/(-1+sqrt(3)) = (3+sqrt(3))/sqrt(2).
Equals 3*A145439.
Equals Gamma(1/24)*Gamma(11/24)/(Gamma(5/24)*Gamma(7/24)) [Zucker] - R. J. Mathar, Jun 24 2024

A337301 Triangle read by rows in which row n lists the closest integers to diagonal lengths of regular n-gon with unit edge length, n >= 4.

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 3, 2, 2, 3, 3, 4, 4, 3, 3, 2, 2, 3, 3, 4, 4, 4, 3, 3, 2, 2, 3, 3, 4, 4, 4, 4, 3, 3, 2, 2, 3, 4, 4, 4, 4, 4, 4, 4, 3, 2, 2, 3, 4, 4, 5, 5, 5, 5, 4, 4, 3, 2, 2, 3, 4, 4, 5, 5, 5, 5, 5, 4, 4, 3, 2

Offset: 4

Mohammed Yaseen, Aug 22 2020

			Triangle begins:
1;
2, 2;
2, 2, 2;
2, 2, 2, 2;
2, 2, 3, 2, 2;
2, 3, 3, 3, 3, 2;
2, 3, 3, 3, 3, 3, 2;
2, 3, 3, 4, 4, 3, 3, 2;
2, 3, 3, 4, 4, 4, 3, 3, 2;
2, 3, 3, 4, 4, 4, 4, 3, 3, 2;
2, 3, 4, 4, 4, 4, 4, 4, 4, 3, 2;
2, 3, 4, 4, 5, 5, 5, 5, 4, 4, 3, 2;
2, 3, 4, 4, 5, 5, 5, 5, 5, 4, 4, 3, 2;
...
Row n lists the closest integers to the length of the diagonals drawn from a fixed vertex of a regular n-gon with unit edge length, n >= 4.
The lengths of the diagonals drawn from vertex A of a regular 8-gon ABCDEFGH with unit edge length are:
AC = 1.84775...
AD = 2.41421...
AE = 2.61312...
AF = 2.41421...
AG = 1.84775...
So the row for n=8 is 2, 2, 3, 2, 2.

Cf. A064313.
Decimal expansion of diagonal lengths of regular n-gons with unit edge length:
n=4  A002193.
n=5  A001622.
n=6  A002194, A000038.
n=7  A160389, A231187.
n=8  A179260, A014176, A121601.
n=9  A332437.
n=10 A188593, A104457, A019970, A134945.
n=11 A231186.
n=12 A188887, A090388, A337402, A019973, A214726.

T[n_,k_]:=Round[Sin[(k+1)*Pi/n]/Sin[Pi/n]]; Flatten[Table[T[n,k],{n,4,16},{k,1,n-3}]] (* Stefano Spezia, Sep 07 2020 *)

T(n,k) = round(sin((k+1)*Pi/n)/sin(Pi/n)), n >= 4, 1 <= k <= n-3.

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A188887 Decimal expansion of sqrt(2 + sqrt(3)).

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A246726 Decimal expansion of r_4, the 4th smallest radius < 1 for which a compact packing of the plane exists, with disks of radius 1 and r_4.

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A337402 Decimal expansion of the length of third shortest diagonal in a regular 12-gon with unit edge length.

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A337301 Triangle read by rows in which row n lists the closest integers to diagonal lengths of regular n-gon with unit edge length, n >= 4.

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