A010503 -id:A010503 - cp's OEIS frontend

A371467 Decimal expansion of Product_{k>=0} (1 - 1/(3*k+2)^2).

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

Offset: 0

Ilya Gutkovskiy, Mar 31 2024

			0.6844634059797257270110769788663463289556838...

Cf. A003881, A010503, A086089, A096427, A224268, A224273, A371466.

RealDigits[(4/3) Pi^2/Gamma[1/3]^3, 10, 100][[1]]

Equals (4/3) * Pi^2 / Gamma(1/3)^3.

Equals 1/A224273. - Hugo Pfoertner, Mar 31 2024

A381485 Decimal expansion of sqrt(13)/6.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Feb 24 2025

The greatest possible minimum distance between 6 points in a unit square.

The solution was found by Ronald L. Graham and reported by Schaer (1965).

			0.60092521257733154885320354457841599104188276230754...

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.

J. Schaer, The densest packing of 9 circles in a square, Canadian Mathematical Bulletin, Vol. 8, No. 3 (1965), pp. 273-277.
D. Würtz, M. Monagan, and R. Peikert, The history of packing circles in a square, Maple Technical Newsletter, Vol. 1 (1994), pp. 35-42; ResearchGate link.
Index entries for algebraic numbers, degree 2.

Solutions for k points: A002193 (k = 2), A120683 (k = 3), 1 (k = 4), A010503 (k = 5), this constant (k = 6), A379338 (k = 7), A101263 (k = 8), A020761 (k = 9), A281065 (k = 10).

Cf. A010470, A142464, A176019, A295330, A344069.

Mathematica
```
RealDigits[Sqrt[13] / 6, 10, 120][[1]]
```

list(len) = digits(floor(10^len*quadgen(52)/6));

Equals A010470 / 6 = A295330 / 3 = A344069 / 2 = A176019 - 1/2 = sqrt(A142464).

Minimal polynomial: 36*x^2 - 13.

A385694 Decimal expansion of the volume of a triaugmented hexagonal prism with unit edge.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jul 07 2025

The triaugmented hexagonal prism is Johnson solid J_57.

			3.3051829925398634646920138743636575896990438184040...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Triaugmented hexagonal prism.

Cf. A385259 (surface area + 7).

Cf. A010503, A104956, A385569, A385578.

First[RealDigits[1/Sqrt[2] + 3*Sqrt[3]/2, 10, 100]]
First[RealDigits[PolyhedronData["J57", "Volume"], 10, 100]]

Equals 1/sqrt(2) + 3*sqrt(3)/2 = A010503 + A104956.

Equals the largest root of 16*x^4 - 232*x^2 + 625.

A079059 Decimal expansion of product( p == 3 (mod 4), sqrt(1-p^-2)).

Original entry on oeis.org

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

Offset: 0

Benoit Cloitre, Feb 02 2003

The complementary product_{p == 1 (mod 4)} sqrt(1-1/p^2) = 0.97303... is related: 0.925261....*0.97303... = sqrt(4/3)/sqrt(Zeta(2)) = 10*A020832/sqrt(A013661). [R. J. Mathar, Jan 31 2009]

E. Landau, "Handbuch der Lehre von der Verteilung der Primzahlen", vol. 2, Teubner, Leipzig; third edition: Chelsea, New York (1974), pp. 641-669

E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, vol. 2, Leipzig, Berlin, B. G. Teubner, 1909.
E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, vol. 1 and vol. 2, Leipzig, Berlin, B. G. Teubner, 1909.

Cf. A071903.

prod(k=1,40000,if(prime(k)%4-3,1,sqrt(1-prime(k)^-2)))

product( p == 3 (mod 4), sqrt(1-p^-2)) = 0.92526...

Equals 1/(sqrt(2)*A064533) = A010503/A064533. [R. J. Mathar, Jul 29 2010]

Corrected offset and leading zero R. J. Mathar, Jan 31 2009
More digits from R. J. Mathar, Jul 28 2010
More digits, using the Jul 29 2010 formula from R. J. Mathar, from Jon E. Schoenfield, Nov 05 2016

A199861 The decimal expansion (unsigned) of the value of d that maximizes the Brahmagupta expression given below.

Original entry on oeis.org

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

Offset: 0

Frank M Jackson, Nov 11 2011

Brahmagupta expression sqrt((-1+1/(1+d)+1/(1+2d)+1/(1+3d)) * (1-1/(1+d)+1/(1+2d)+1/(1+3d)) * (1+1/(1+d)-1/(1+2d)+1/(1+3d)) * (1+1/(1+d)+1/(1+2d)-1/(1+3d)))/4 for d in the interval [-1/3, inf] where 1/(1+d), 1/(1+2d) and 1/(1+3d) are always positive.

The area of a convex quadrilateral with fixed sides is maximal when it is organized as a convex cyclic quadrilateral. Furthermore in order that a quadrilateral can have sides in a harmonic progression 1 : 1/(1+d) : 1/(1+2d) : 1/(1+3d) its denominator's common difference d is limited to the range f < d < g where f is the constant A199590 and g is the constant A199589. Consequently when d=-0.2271064482... it maximizes Brahmagupta's expression for the area of a convex cyclic quadrilateral whose sides form a harmonic progression.

			-0.22710644829438120301114335253234461837754...

Wikipedia, Brahmagupta's formula.
Index entries for algebraic numbers, degree 12

Cf. A010503, A193180, A193211, A199589, A199590.

RealDigits[d/.NMaximize[{Sqrt[(-1+1/(1+d)+1/(1+2d)+1/(1+3d))(1-1/(1+d)+1/(1+2d)+1/(1+3d))(1+1/(1+d)-1/(1+2d)+1/(1+3d))(1+1/(1+d)+1/(1+2d)-1/(1+3d))]/4, -1/4120, PrecisionGoal->100, WorkingPrecision->240][[2]]][[1]]

real(polroots(1323*d^12 + 9711*d^11 + 32535*d^10 + 67005*d^9 + 94338*d^8 + 94761*d^7 + 68955*d^6 + 36367*d^5 + 13740*d^4 + 3619*d^3 + 630*d^2 + 65*d + 3)[4]) \\ Charles R Greathouse IV, Nov 11 2011

polrootsreal(1323*x^12 - 9711*x^11 + 32535*x^10 - 67005*x^9 + 94338*x^8 - 94761*x^7 + 68955*x^6 - 36367*x^5 + 13740*x^4 - 3619*x^3 + 630*x^2 - 65*x + 3)[1] \\ Charles R Greathouse IV, Oct 27 2023

d is the largest real root of the equation 1323d^12 + 9711d^11 + 32535d^10 + 67005d^9 + 94338d^8 + 94761d^7 + 68955d^6 + 36367d^5 + 13740d^4 + 3619d^3 + 630d^2 + 65d + 3 = 0.

A318614 Scaled g.f. S(u) = Sum_{n>0} a(n)16(u/16)^n satisfies T(u) = d/du S(u), with T(u) as defined by A318417; sequence gives a(n).

Original entry on oeis.org

1, 6, 76, 1260, 24276, 515592, 11721072, 280020312, 6945369860, 177358000248, 4635276570288, 123449340098448, 3339525750984528, 91535631253610400, 2537277723600799680, 71015600640006437040, 2004523477053308685540, 57003431104378084982040

Offset: 1

Bradley Klee, Aug 30 2018

nonn

Area interior to the central loop of u = 2*H = x^2 + y^2 - (1/2)*(x^4 + y^4) equals to Pi*S(u), when u in [0,1/2].

			Singular Value: S(1/2) = 1/sqrt(2).
N=4, h=1/sqrt(2) Quantization: S(u) = (n+1/2)*h/N.
  n  |                  u
==================================================
  0  |  0.08544689553344134756293807606337...
  1  |  0.23840989875904155311088418238272...
  2  |  0.36638282702449450473835851051425...
  3  |  0.46595506694324457665483887176081...

E. Heller, The Semiclassical Way to Dynamics and Spectroscopy, Princeton University Press, 2018, page 204.

Eric Weisstein's World of Mathematics, Plane Division by Ellipses.
Eric Weisstein's World of Mathematics, Circle Ellipse Intersection.

Cf. A318417, A010503, A247719, A000888.

a:=[1,6];; for n in [3..20] do a[n]:=(1/(n*(n-1)^2))*(12*(n-1)*(2*n-3)^2*a[n-1]-(128*(n-2)*(2*n-5)*(2*n-3)*a[n-2])); od; a; # Muniru A Asiru, Sep 24 2018

RecurrenceTable[{(n-1)^2*n*a[n] - 12*(n-1)*(2*n-3)^2*a[n-1] + 128*(n-2)*(2*n-5)*(2*n-3)*a[n-2] == 0, a[1] == 1, a[2] == 6}, a, {n, 1, 1000}]

(n-1)^2*n*a(n) - 12*(n-1)*(2*n-3)^2*a(n-1) + 128*(n-2)*(2*n-5)*(2*n-3)*a(n-2) == 0.

a(n) = A000108(n-1)*A098410(n-1).

A322505 Factorial expansion of 1/sqrt(2) = Sum_{n>=1} a(n)/n!.

Original entry on oeis.org

0, 1, 1, 0, 4, 5, 0, 6, 4, 9, 0, 11, 7, 3, 11, 10, 2, 2, 5, 16, 11, 3, 7, 18, 16, 19, 11, 12, 21, 19, 22, 5, 31, 21, 25, 30, 20, 6, 5, 21, 17, 41, 36, 14, 28, 13, 45, 16, 0, 33, 1, 2, 41, 1, 28, 43, 9, 15, 16, 28, 22, 19, 22, 13, 34, 61, 38, 40, 56, 44, 69, 25, 42, 44, 34, 73, 71, 42, 17

Offset: 1

G. C. Greubel, Dec 12 2018

nonn

			1/sqrt(2) = 0 + 1/2! + 1/3! + 0/4! + 4/5! + 5/6! + 0/7! + 6/8! + ...

Index entries for factorial base representation

Cf. A010503 (decimal expansion), A130130 (continued fraction).

Cf. A009949 (sqrt(2)).

SetDefaultRealField(RealField(250));  [Floor(1/Sqrt(2))] cat [Floor(Factorial(n)/Sqrt(2)) - n*Floor(Factorial((n-1))/Sqrt(2)) : n in [2..80]];

With[{b = 1/Sqrt[2]}, Table[If[n == 1, Floor[b], Floor[n!*b] - n*Floor[(n - 1)!*b]], {n, 1, 100}]] (* G. C. Greubel, Dec 12 2018 *)

default(realprecision, 250); b = 1/sqrt(2); for(n=1, 80, print1(if(n==1, floor(b), floor(n!*b) - n*floor((n-1)!*b)), ", "))

b=1/sqrt(2);
def a(n):
    if (n==1): return floor(b)
    else: return expand(floor(factorial(n)*b) -n*floor(factorial(n-1)*b))
[a(n) for n in (1..80)]

A351898 Decimal expansion of metallic ratio for N = 14.

Original entry on oeis.org

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

Offset: 2

A.H.M. Smeets, Feb 24 2022

Decimal expansion of continued fraction [14; 14, 14, 14, ...].
Also largest solution of x^2 - 14 x - 1 = 0.
Essentially the same digit sequence as A010503, A157214, A174968 and A268683.
The metallic ratio's for N = A077444(n) are equal to powers of the silver ratio, i.e., A014166^(2n-1); this constant represents the special case for N = A077444(2).

			14.0710678118654752440084436210484903928483593...

Michael Penn, 7+5√2 is a perfect cube??, YouTube video, 2022.
Wikipedia, Metallic mean.

Cf. A010503, A077444, A157214, A174968, A268683.

Metallic ratios: A001622 (N=1), A014176 (N=2), A098316 (N=3), A098317 (N=4), A098318 (N=5), A176398 (N=6), A176439 (N=7), A176458 (N=8), A176522 (N=9), A176537 (N=10), A244593 (N=11).

RealDigits[7 + 5*Sqrt[2], 10, 100][[1]] (* Amiram Eldar, Feb 24 2022 *)

PARI
```
(1+sqrt(2))^3
```

Equals 2 + 5*A014176.
Equals A014176^3.
Equals exp(arcsinh(7)). - Amiram Eldar, Jul 04 2023

A358614 Decimal expansion of 9*sqrt(2)/32.

Original entry on oeis.org

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

Offset: 0

Bernard Schott, Dec 05 2022

Smallest constant M such that the inequality
|a*b*(a^2 - b^2) + b*c*(b^2 - c^2) + c*a*(c^2 - a^2)| <= M * (a^2 + b^2 + c^2)^2
holds for all real numbers a, b, c.
Equality stands for any triple (a, b, c) proportional to (1 - 3*sqrt(2)/2, 1, 1 + 3*sqrt(2)/2), up to permutation.
This constant is the answer to the 3rd problem, proposed by Ireland during the 47th International Mathematical Olympiad in 2006 at Ljubljana, Slovenia (see links).
Equivalently |(a - b)(b - c)(c - a)(a + b + c)| / (a^2 + b^2 + c^2)^2 <= M  with (a,b,c) != (0,0,0).

			0.3977475644174329824...

Evan Chen, IMO 2006/3, IMO 2006 Solution Notes.
The IMO compendium, Problem 3, 47th IMO 2006.
Index to sequences related to Olympiads.

Cf. A002193, A010474, A010503, A230981.

Maple
```
evalf(9*sqrt(2)/32), 100);
```

RealDigits[9*Sqrt[2]/32, 10, 120][[1]] (* Amiram Eldar, Dec 05 2022 *)

Equals (3/16) * A230981 = (3/32) * A010474 = (9/32) * A002193 = (9/16) * A010503.

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

cp's OEIS Frontend

A371467 Decimal expansion of Product_{k>=0} (1 - 1/(3*k+2)^2).

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A381485 Decimal expansion of sqrt(13)/6.

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A385694 Decimal expansion of the volume of a triaugmented hexagonal prism with unit edge.

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A079059 Decimal expansion of product( p == 3 (mod 4), sqrt(1-p^-2)).

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Extensions

A199861 The decimal expansion (unsigned) of the value of d that maximizes the Brahmagupta expression given below.

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A318614 Scaled g.f. S(u) = Sum_{n>0} a(n)*16*(u/16)^n satisfies T(u) = d/du S(u), with T(u) as defined by A318417; sequence gives a(n).

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A322505 Factorial expansion of 1/sqrt(2) = Sum_{n>=1} a(n)/n!.

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A318614 Scaled g.f. S(u) = Sum_{n>0} a(n)16(u/16)^n satisfies T(u) = d/du S(u), with T(u) as defined by A318417; sequence gives a(n).