A010527 -id:A010527 - cp's OEIS frontend

A093822 Decimal expansion of -109/121 - 82/(121sqrt(3)) + (2sqrt(-35139 + 28634*sqrt(3)))/121 - Pi/3 + arccos((-1 + sqrt(3))/2).

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

Offset: 0

Eric W. Weisstein, Apr 16 2004

Area of lamina found by Sprague in the Lebesgue minimal problem.

			0.844137123...

Eric Weisstein's World of Mathematics, Lebesgue Minimal Problem

Cf. A010527, A093821.

RealDigits[-109/121-82/(121Sqrt[3])+(2Sqrt[-35139+28634Sqrt[3]])/121-Pi/3+ ArcCos[(-1+Sqrt[3])/2],10,120][[1]] (* Harvey P. Dale, Sep 22 2020 *)

A194082 Sum{floor(sqrt(3)*k/2) : 1<=k<=n}.

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 21, 27, 34, 42, 51, 61, 72, 84, 96, 109, 123, 138, 154, 171, 189, 208, 227, 247, 268, 290, 313, 337, 362, 387, 413, 440, 468, 497, 527, 558, 590, 622, 655, 689, 724, 760, 797, 835, 873, 912, 952, 993, 1035, 1078, 1122, 1167, 1212

Offset: 1

Clark Kimberling, Aug 17 2011

nonn

Partial sums of A171970.
Comment from R. J. Mathar, Dec 02 2012 (Start):
a(n-1) is the number of unit squares regularly packed into the isosceles triangle of edge length n.
The triangle may be aligned with the Cartesian axes by putting its bottom edge on the horizontal axis, so its vertices are at (x,y) = (0,0), (n,0) and (n/2,sqrt(3)*n/2), see A010527.
The area inside the triangle is sqrt(3)*n^2/4 = A120011*n^2. There is an obvious upper limit of floor(sqrt(3)*n^2/4) = A171971(n) to the count of non-overlapping unit squares inside this triangle.
Regular packing: We place the first row of unit squares so they touch the bottom edge of the triangle. Their number is limited by the length of the horizontal section of the line y=1 inside the triangle, n-2*y/sqrt(3), which touches all of these first-row squares at their top.
The number of unit squares in the next row, between y=1 and y=2, is limited by the length of the horizontal section of the line y=2 inside the triangle, n-2*y/sqrt(3). Continuing, in row y=1, 2, ... we insert floor(n-2*y/sqrt(3)) unit squares, all with the same orientation.
The total number of squares is sum_{ y=1, 2, ..., floor(n*sqrt(3)/2) } floor( n-2*y/sqrt(3) ), and resummation yields, up to an index shift, this sequence here.
(End)

Cf. A171970.

r = Sqrt[3]/2;
c[k_] := Sum[Floor[j*r], {j, 1, k}];
Table[c[k], {k, 1, 90}]

a(n)=sum(k=1,n,sqrtint(3*k^2\4)) \\ Charles R Greathouse IV, Jan 06 2013

A333322 Decimal expansion of (3/8) * sqrt(3).

Original entry on oeis.org

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

Offset: 0

Kritsada Moomuang, Mar 15 2020

This is the area of the regular hexagon of diameter 1.
From Bernard Schott, Apr 09 2022 and Oct 01 2022: (Start)
For any triangle ABC, where (A,B,C) are the angles:
  sin(A) * sin(B) * sin(C) <= (3/8) * sqrt(3) [Bottema reference],
  cos(A/2) * cos(B/2) * cos(C/2) <= (3/8) * sqrt(3) [Mitrinovic reference],
and if (ha,hb,hc) are the altitude lengths and (a,b,c) the side lengths of this triangle [Scott Brown link]:
  (ha+hb) * (hb+hc) * (hc+ha) / (a+b) * (b+c) * (c+a) <= (3/8) * sqrt(3).
The equalities are obtained only when triangle ABC is equilateral. (End)

			0.649519052838328985...

O. Bottema et al., Geometric Inequalities, Groningen, 1969, item 2.7, page 19.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 207.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.15, p. 526.
D. S. Mitrinovic, E. S. Barnes, D. C. B. Marsh, J. R. M. Radok, Elementary Inequalities, Tutorial Text 1 (1964), P. Noordhoff LTD, Groningen, problem 6.2.2, page 111.

Scott Brown, Problem 3453, Crux Mathematicorum, Vol. 36, No. 5 (2010), pp. 342 and 343.

Cf. A002194 (sqrt(3)), A104954.

Cf. A010527, A020821, A104956, A152623 (other geometric inequalities).

RealDigits[(3/8) * Sqrt[3], 10, 120][[1]]

sqrt(27)/8 \\ Charles R Greathouse IV, Apr 09 2022

Equals A104954/2 or A104956/4.

A179453 Decimal expansion of the inradius of an icosidodecahedron with edge length 1.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 14 2010

Icosidodecahedron: 32 faces, 30 vertices, and 60 edges.

			1.46352549156242113615344012577422858829023188485432214660158646702894...

Icosidodecahedron

Cf. A001622, A010527, A102208, A179290, A179292, A179294, A179449, A179450, A179451, A179452.

Mathematica
```
RealDigits[N[(5+3*Sqrt[5])/8,200]]
```

(sqrt(45)+5)/8 \\ Charles R Greathouse IV, Apr 25 2016

Digits of (5+3*sqrt(5))/8.

A240198 Numerator of rational approximation of sqrt(3)/2 within a tolerance 10^(-n).

Original entry on oeis.org

0, 4, 6, 13, 84, 181, 989, 2521, 11254, 35113, 35113

Offset: 0

Zak Seidov, Apr 02 2014

nonn

Rational approximations of sqrt(3)/2 are 0/1, 4/5, 6/7, 13/15, 84/97, 181/209, 989/1142, 2521/2911, 11254/12995, 35113/40545, 35113/40545.

Cf. A010527.

Table[Numerator[Rationalize[Sqrt[3]/2, 10^(-n)]], {n, 0, 10}]

A272526 Decimal expansion of s_4, a 4-dimensional Steiner ratio analog.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, May 02 2016

			0.7439856178281340629943798859204105522737599475964283917092969185...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 8.6 Steiner Tree Constants, p. 505.

D. Z. Du and W. D. Smith , Disproofs of Generalized Gilbert-Pollak Conjecture on the Steiner Ratio in Three or More Dimensions, Journal of Combinatorial Theory, Series A Volume 74, Issue 1, April 1996, Pages 115-130
Eric Weisstein's MathWorld, Steiner Tree.
Wikipedia, Steiner Tree problem

Cf. A010527, A220351, A248411.

s4 = Root[900 s^8 - 1863 s^6 + 2950 s^4 - 1511 s^2 + 164, s, 4];
RealDigits[s4, 10, 98][[1]]

Minimal polynomial is 900 s^8 - 1863 s^6 + 2950 s^4 - 1511 s^2 + 164.

A354249 Decimal expansion of 27sqrt(3) / (2Pi).

Original entry on oeis.org

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

Offset: 1

Bernard Schott, May 20 2022

For any triangle ABC (see Crux Mathematicorum):
(sin(A) + sin(B) + sin(C)) * (1/A + 1/B + 1/C) >= 27*sqrt(3) / (2*Pi),
where (A,B,C) are the angles in radians.
Equality stands iff triangle ABC is equilateral.

			7.442940088194192668402907727...

Shi-Chang Chi and Ji Chen, Douglas L. Grant, Problem 2015, Crux Mathematicorum, Vol. 22, No. 1 (1996), pp. 47-48.
Index entries for transcendental numbers

Cf. A010527, A132702, A132717.

Maple
```
evalf(27*sqrt(3)/(2*Pi),110);
```

RealDigits[(27*Sqrt[3]/(2*Pi)), 10, 110][[1]] (* Amiram Eldar, May 21 2022; corrected by Georg Fischer, Aug 04 2024 *)

Equals A010527 * A132717.

a(100) corrected by Georg Fischer, Aug 04 2024

A373642 Decimal expansion of Sum_{k>=1} (sin(Pi/k))^(2k).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Jun 12 2024

			0^2 + 1^4 + (0.86602...)^6 + (0.70710..)^8 + (0.58778..)^10 + ... = 1.4895502488138264685841...

Lewis Trem, Infinite sum of ..., Math StackExchange, Jul 8 2020.

Cf. A269611, A010527 (sin Pi/3), A010503 (sin Pi/4), A019845 (sin Pi/5).

sumpos(k = 1, sin(Pi/k)^(2*k)) \\ Amiram Eldar, Aug 20 2024

A375741 Decimal expansion of 6Pi/(3sqrt(3) + 8*Pi).

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Aug 26 2024

This constant expresses the expected proportion of individuals that belong to reciprocal nearest-neighbor relationship pairs in a population of random patterns over a plane.

			0.62150489688743159140596781880802827312707885...

E. C. Pielou, An Introduction to Mathematical Ecology, John Wiley & Sons, Inc. 1969. See pp. 121-122.

Index entries for transcendental numbers.

Cf. A000796, A010482, A010527, A019692, A019699, A228719, A228824.

RealDigits[6Pi/(3Sqrt[3]+8Pi),10,100][[1]]

Equals Integral_{x=0..oo} 2*Pi*x*exp(-x^2*(sqrt(3)/2+4*Pi/3)) dx. [Pielou, 1969]

cp's OEIS Frontend

A093822 Decimal expansion of -109/121 - 82/(121*sqrt(3)) + (2*sqrt(-35139 + 28634*sqrt(3)))/121 - Pi/3 + arccos((-1 + sqrt(3))/2).

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A194082 Sum{floor(sqrt(3)*k/2) : 1<=k<=n}.

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A333322 Decimal expansion of (3/8) * sqrt(3).

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A179453 Decimal expansion of the inradius of an icosidodecahedron with edge length 1.

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A240198 Numerator of rational approximation of sqrt(3)/2 within a tolerance 10^(-n).

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A272526 Decimal expansion of s_4, a 4-dimensional Steiner ratio analog.

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A354249 Decimal expansion of 27*sqrt(3) / (2*Pi).

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A373642 Decimal expansion of Sum_{k>=1} (sin(Pi/k))^(2k).

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A093822 Decimal expansion of -109/121 - 82/(121sqrt(3)) + (2sqrt(-35139 + 28634*sqrt(3)))/121 - Pi/3 + arccos((-1 + sqrt(3))/2).

A354249 Decimal expansion of 27sqrt(3) / (2Pi).

A375741 Decimal expansion of 6Pi/(3sqrt(3) + 8*Pi).