A010466 -id:A010466 - cp's OEIS frontend

A265305 Decimal expansion of Sum_{k>=1} (c(2k)-c(2k-1)), where c = convergents to sqrt(8).

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

Offset: 1

Clark Kimberling, Dec 13 2015

			sum = 1.0343484404413437724399287046773384722110427146999...

Cf. A010466, A265303, A265304, A265288 (guide).

x = Sqrt[8]; z = 600; c = Convergents[x, z];
s1 = Sum[x - c[[2 k - 1]], {k, 1, z/2}]; N[s1, 200]
s2 = Sum[c[[2 k]] - x, {k, 1, z/2}]; N[s2, 200]
N[s1 + s2, 200]
RealDigits[s1, 10, 120][[1]]  (* A265303 *)
RealDigits[s2, 10, 120][[1]]  (* A265304 *)
RealDigits[s1 + s2, 10, 120][[1]](* A265305 *)

A280585 Decimal expansion of 8*sin(Pi/8).

Original entry on oeis.org

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

Offset: 1

Rick L. Shepherd, Jan 05 2017

Decimal expansion of the ratio of the perimeter of a regular 8-gon (octagon) to its diameter (largest diagonal).

			3.061467458920718173827679872243190934090756499885016331470405085020368271...

Index entries for algebraic numbers, degree 4.

Cf. For other n-gons: A010466 (n=4), 10*A019827 (n=5, 10), A280533 (n=7), A280633 (n=9), A280725 (n=11), A280819 (n=12).

Cf. A182168.

evalf(8*sin(Pi/8),100); # Wesley Ivan Hurt, Feb 01 2017

RealDigits[8*Sin[Pi/8], 10, 120][[1]] (* Amiram Eldar, Jun 26 2023 *)

PARI
```
8*sin(Pi/8)
```

Equals 8*A182168.

A280633 Decimal expansion of 18*sin(Pi/18).

Original entry on oeis.org

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

Offset: 1

Rick L. Shepherd, Jan 06 2017

The ratio of the perimeter of a regular 9-gon (nonagon) to its diameter (largest diagonal).

Also least positive root of x^3 - 243x + 729.

			3.125667198004746279330899281847666328006762189313248970252344806377184798...

Index entries for algebraic numbers, degree 3.

Cf. For other n-gons: A010466 (n=4), 10*A019827 (n=5, 10), A280533 (n=7),A280585 (n=8), A280725(n=11), A280819 (n=12).

evalf(18*sin(Pi/18),100); # Wesley Ivan Hurt, Feb 01 2017

RealDigits[18*Sin[Pi/18],10,120][[1]] (* Harvey P. Dale, Dec 02 2018 *)

PARI
```
18*sin(Pi/18)
```

18*A019819.

A280819 Decimal expansion of 12*sin(Pi/12).

Original entry on oeis.org

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

Offset: 1

Rick L. Shepherd, Jan 08 2017

The ratio of the perimeter of a regular 12-gon (dodecagon) to its diameter (greatest diagonal).

A quartic integer: the least positive root of x^4 - 144x^2 + 1296.

			3.105828541230249148186786051488579940188826815839166165768038487780683696...

Index entries for algebraic numbers, degree 4

Cf. For other n-gons: A010466 (n=4), 10*A019827 (n=5, 10), A280533 (n=7), A280585 (n=8), A280633 (n=9), A280725 (n=11).

First@ RealDigits@ N[12 Sin[Pi/12], 120] (* Michael De Vlieger, Feb 05 2017 *)

PARI
```
12*sin(Pi/12)
```

polrootsreal(x^4-144*x^2+1296)[3] \\ Charles R Greathouse IV, Feb 05 2017

12*A019824.

A305308 Decimal expansion of Lagrange(4) = sqrt(1517)/13.

Original entry on oeis.org

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

Offset: 1

Wolfdieter Lang, Jun 25 2018

For every irrational number alpha not equivalent to each of the following three numbers i) golden section A001622, ii) sqrt(2) = A002193 and iii) (5 + sqrt(221))/14 = A177841 there exist infinitely many rational numbers h/k (in lowest terms) such that |alpha - h/k| < 1/(Lagrange(4)*k^2). The constant L(4) cannot be replaced by a larger number because then the statement becomes false for, e.g., alpha = (23 + sqrt(1517))/26. Two real numbers x and y are equivalent if there exist integers p, q, r and s with |p*s - q*r| = 1 such that y = (p*x + q)/(r*x + s) (unimodular transformation). This means that the continued fractions of x and y become eventuakky identical.
See the references (in Havil, p. 174, equivalence classes of numbers should have been excluded).
The continued fraction of Lagrange(4) is [2; repeat(1, 252, 3, 1012, 3, 252, 1, 4)]. 1/L(4) = 0.3337725078... < 1/3.
Perron's numbers M(xi) (pp. 4, 5), for M(xi) < 3, are the Lagrange numbers sqrt(9*Q^2 - 4)/Q, with Q = Q(n) = A002559(n), n >= 1, and his corresponding xi(4) = (sqrt(1517) + 23)/26 with a purely periodic simple continued fraction [repeat(2, 2, 1, 1, 1, 1)].
Cassels (p. 18) uses the version: For irrational theta not equivalent to the above given three numbers i), ii) and iii) there are infinitely many solutions of q*||q*theta|| < 1/Lagrange(4), where 1/Lagrange(4) cannot be improved for theta equivalent to -29/26 + (1/26)*sqrt(1517). Here ||x|| is the positive difference between x and the nearest integer.

			2.9960526298692994692341394026263186397583021915005644481405263406560103404...

J. W. S. Cassels, An Introduction to Diophantine Approximation, Cambridge University Press, 1957, Chapter II, The Markoff Chain, pp. 18-44.
Julian Havil, The Irrationals, Princeton University Press, Princeton and Oxford, 2012, pp. 174-175 and 221-224.
J. F. Koksma, Diophantische Approximationen, 1936, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vierter Band, Teil 4, Julius Springer, Berlin, pp. 29-33.
Ivan Niven, Diophantine Approximations, Interscience Publishers, 1963, p. 14.
Oskar Perron, Über die Approximation irrationaler Zahlen durch rationale, 4. Abhandlung, pp. 1- 17, and part II., 8. Abhandlung, pp. 1-12. Sitzungsber. Heidelberger Akademie der Wiss., 1921, Carls Winters Universitätsbuchhandlung.
Paulo Ribenboim, Meine Zahlen, meine Freunde, 2009, Springer, 10. 6 B, pp. 312-314.
Jörn Steuding, Diophantine Analysis, 2005, Chapman & Hall/CRC, pp. 80-82.

Encyclopedia of Mathematics, Diophantine approximations.
A. A. Markoff, Sur les formes quadratiques binaires indéfinies, Math. Ann, 15 (18) (1879) 381-406.
A. A. Markoff, Sur les formes quadratiques binaires indéfinies (Second mémoire), Math. Ann, 17 (1880) 379-399.

The Lagrange numbers for n = 1..3 are A002163, A010466, A200991.

Cf. A001622, A002559.

RealDigits[Sqrt[1517]/13,10,120][[1]] (* Harvey P. Dale, Apr 12 2022 *)

Lagrange(4) = sqrt(9*M(4)^2 - 4)/M(4) = sqrt(9*13^2 - 4)/13 = sqrt(1517)/13, with the Markoff number M(4) = A002559(4) = 13.

A378128 Decimal expansion of 2/L, where L is the lemniscate constant (A062539).

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Nov 17 2024

			0.76275976350181318806232598096361579329262923734807...

Wikipedia, Lemniscate constant (includes this constant).
Index to sequences related to the Gamma function.

Cf. A010466, A062539, A085565, A175575.

Cf. A377999, A378129, A378130, A378131, A378132.

First[RealDigits[Sqrt[8]*Gamma[3/4]^2/Pi^(3/2), 10, 100]]

Equals 1/A085565.
Equals 2*sqrt(2)*Gamma(3/4)^2/Pi^(3/2) = A010466*A175575.
Equals Product_{k >= 1} b(k), where b(1) = sqrt(1/2) and, for k >= 2, b(k) = sqrt(1/2 + (1/2)/b(k-1)).

A384139 Decimal expansion of the volume of an elongated triangular bipyramid with unit edges.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, May 20 2025

The elongated triangular bipyramid is Johnson solid J_14.

Also the volume of an augmented triangular prism (Johnson solid J_49) with unit edges. - Paolo Xausa, Aug 04 2025

			0.66871496228773516484880970607808443816397995934875...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Wikipedia, Augmented triangular prism.
Wikipedia, Elongated triangular bipyramid.

Cf. A165663 (surface area - 2).

Cf. A010482, A010482, A383852, A384138, A384140.

First[RealDigits[(Sqrt[8] + Sqrt[27])/12, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J14", "Volume"], 10, 100]]

Equals (2*sqrt(2) + 3*sqrt(3))/12 = (A010466 + A010482)/12.

Equals the largest root of 20736*x^4 - 10080*x^2 + 361.

A386002 Decimal expansion of the volume of an augmented tridiminished icosahedron with unit edge.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jul 18 2025

The augmented tridiminished icosahedron is Johnson solid J_64.

			1.3950376236351965821861497137307637418843196778...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Augmented tridiminished icosahedron.

Cf. A386003 (surface area).

Cf. A002163, A010466, A020829, A102208, A179552, A385857, A386000.

First[RealDigits[(15 + Sqrt[8] + 7*Sqrt[5])/24, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J64", "Volume"], 10, 100]]

Equals (15 + 2*sqrt(2) + 7*sqrt(5))/24 = (15 + A010466 + 7*A002163)/24.
Equals A102208 - 3*A179552 + A020829 = A386000 + A020829.
Equals the largest root of 2304*x^4 - 5760*x^3 + 3376*x^2 + 280*x - 49.

A386411 Decimal expansion of the volume of an augmented truncated tetrahedron with unit edge.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jul 21 2025

The augmented truncated tetrahedron is Johnson solid J_65.

			3.889087296526011384204643991576669716066597657...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Augmented truncated tetrahedron.

Cf. A386412 (surface area).

Cf. A010466, A020829, A377275.

First[RealDigits[11/Sqrt[8], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J65", "Volume"], 10, 100]]

Equals 11/(2*sqrt(2)) = 11/A010466.
Equals A377275 + 10*A020829.
Equals the largest root of 8*x^2 - 121.

A144982 Decimal expansion of cos(Pi/24) = cos(7.5 degrees).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Sep 28 2008

Octic number of denominator 2 and minimal polynomial 256x^8 - 512x^6 + 320x^4 - 64x^2 + 1. - Charles R Greathouse IV, May 13 2019

			Equals 0.9914448613738104111445575269285628712777382744...

Index entries for algebraic numbers, degree 8

evalf( sqrt(2*sqrt(2)+sqrt(3)+1)/2^(5/4)) ;

RealDigits[ Sqrt[2 + Sqrt[2 + Sqrt[3]]]/2, 10, 105] // First (* Jean-François Alcover, Feb 20 2013 *)
RealDigits[Cos[Pi/24],10,120][[1]] (* Harvey P. Dale, Feb 11 2024 *)

cos(Pi/24) \\ Charles R Greathouse IV, May 13 2019

sqrt(2*sqrt(2)+sqrt(3)+1)/2^(5/4) =sqrt(A010466+A090388)/A011027.
Equals 2F1(9/16,7/16;1/2;3/4) / 2 . - R. J. Mathar, Oct 27 2008
4*this^3 -3*this = A144981. - R. J. Mathar, Aug 29 2025
Equals 2F1(-1/16,1/16;1/2;3/4) = 2F1(-1/12,1/12;1/2;1/2). - R. J. Mathar, Aug 31 2025

cp's OEIS Frontend

A265305 Decimal expansion of Sum_{k>=1} (c(2k)-c(2k-1)), where c = convergents to sqrt(8).

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A280585 Decimal expansion of 8*sin(Pi/8).

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A280633 Decimal expansion of 18*sin(Pi/18).

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A280819 Decimal expansion of 12*sin(Pi/12).

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A305308 Decimal expansion of Lagrange(4) = sqrt(1517)/13.

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A378128 Decimal expansion of 2/L, where L is the lemniscate constant (A062539).

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A384139 Decimal expansion of the volume of an elongated triangular bipyramid with unit edges.

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