A329247 -id:A329247 - 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)

A101263 Decimal expansion of sqrt(2 - sqrt(3)), edge length of a regular dodecagon with circumradius 1.

Original entry on oeis.org

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

Offset: 0

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Jan 25 2005

sqrt(2 - sqrt(3)) is the shape of the lesser sqrt(6)-contraction rectangle, as defined at A188739. - Clark Kimberling, Apr 16 2011
This is a constructible number, since 12-gon is a constructible polygon. See A003401 for more details. - Stanislav Sykora, May 02 2016
It is also smaller positive coordinate of (symmetrical) intersection points of x^2 + y^2 = 4 circle and y = 1/x hyperbola. The bigger coordinate is A188887. - Leszek Lezniak, Sep 18 2018
The greatest possible minimum distance between 8 points in a unit square (Schaer and Meir, 1965; Schaer, 1965; Croft et al., 1991). - Amiram Eldar, Feb 24 2025

			0.517638090205041524697797675248096656698137802639861027628006414630113....

Hallard T. Croft, Kenneth J. Falconer, and Richard K. Guy, Unsolved Problems in Geometry, Springer, 1991, Section D1, p. 108.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.2, p. 487.

G. C. Greubel, Table of n, a(n) for n = 0..5000
J. Schaer and A. Meir, On a geometric extremum problem, Canadian Mathematical Bulletin, Vol. 8, No. 1 (1965), pp. 21-27.
J. Schaer, The densest packing of 9 circles in a square, Canadian Mathematical Bulletin, Vol. 8, No. 3 (1965), pp. 273-277.
Eric Weisstein's World of Mathematics, Dodecagon.
Index entries for algebraic numbers, degree 4.

Cf. A003401, A019824, A019913, A091999, A120683, A188739, A188887, A329247.

r = 6^(1/2); t = (r - (-4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]  (*A101263*)
RealDigits[Sqrt[2-Sqrt[3]],10,120][[1]] (* Harvey P. Dale, Apr 24 2018 *)

2*sin(Pi/12) \\ Stanislav Sykora, May 02 2016

Equals sqrt(A019913). - R. J. Mathar, Apr 20 2009
Equals 2*sin(Pi/12) = 2*cos(Pi*5/12). - Stanislav Sykora, May 02 2016
Equals i^(5/6) + i^(-5/6). - Gary W. Adamson, Jul 07 2022
From Amiram Eldar, Nov 24 2024: (Start)
Equals A120683 / 2 = 2 * A019824 = 1 / A188887 = exp(-A329247).
Equals (sqrt(3)-1)/sqrt(2).
Equals Product_{k>=1} (1 + (-1)^k/A091999(k)). (End)

A329246 Decimal expansion of Sum_{k>=1} cos(k*Pi/4)/k.

Original entry on oeis.org

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

Offset: 0

Jianing Song, Nov 09 2019

Sum_{k>=1} cos(k*x)/k = Re(Sum_{k>=1} exp(k*x*i)/k) = Re(-log(1-exp(x*i))) = -log(2*|sin(x/2)|), x != 2*m*Pi, where i is the imaginary unit.
In general, for real s and complex z, let f(s,z) = Sum_{k>=1} z^k/k^s, then:
(a) if s <= 0, then f(s,z) converges to Polylog(s,z) if |z| < 1;
(b) if 0 < s <= 1, then f(s,z) converges to Polylog(s,z) if z != 1;
(c) if s > 1, then f(s,z) converges to Polylog(s,z) if |z| <= 1.
As a result, let z = e^(i*x), then the series Sum_{k>=1} (cos(k*x) + i*sin(k*x))/k^s converges to Polylog(s,e^(i*x)) if and only if s > 1, or 0 < s <= 1 and x != 2*m*Pi.

			Sum_{k>=1} cos(k*Pi/4)/k = -log(2*|sin(Pi/8)|) = 0.2673999983...

Similar sequences:
A263192 (Sum_{k>=1} cos(k)/sqrt(k) = Re(Polylog(1/2,exp(i))));
A263193 (Sum_{k>=1} sin(k)/sqrt(k) = Im(Polylog(1/2,exp(i))));
A329247 (Sum_{k>=1} cos(k*Pi/6)/k = Re(Polylog(1,exp(i*Pi/6))));
A121225 (Sum_{k>=1} cos(k)/k = Re(Polylog(1,exp(i))));
this sequence (Sum_{k>=1} cos(k*Pi/4)/k = Re(Polylog(1,exp(i*Pi/4))));
A096444 (Sum_{k>=1} sin(k)/k = Im(Polylog(1,exp(i))));
A122143 (Sum_{k>=1} cos(k)/k^2 = Re(Polylog(2,exp(i))));
A096418 (Sum_{k>=1} sin(k)/k^2 = Im(Polylog(2,exp(i)))).

RealDigits[Log[1 + Sqrt[2]/2]/2, 10, 120][[1]] (* Amiram Eldar, May 31 2023 *)

default(realprecision, 100); log(1 + sqrt(2)/2)/2

Equals log(1 + sqrt(2)/2)/2.

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

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A101263 Decimal expansion of sqrt(2 - sqrt(3)), edge length of a regular dodecagon with circumradius 1.

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A329246 Decimal expansion of Sum_{k>=1} cos(k*Pi/4)/k.

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