A010503 -id:A010503 - cp's OEIS frontend

A374957 Decimal expansion of the circumradius of a regular heptagon with unit side length.

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

Offset: 1

Paolo Xausa, Jul 26 2024

			1.15238243548124325262057511177342755672225094380...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Regular Polygon.
Wikipedia, Regular polygon.
Index entries for algebraic numbers, degree 6.

Cf. A374971 (apothem), A374972 (sagitta), A178817 (area).
Cf. circumradius of other polygons with unit side length: A020760 (triangle), A010503 (square), A300074 (pentagon), A285871 (octagon), A375151 (9-gon), A001622 (10-gon), A375190 (11-gon), A188887 (12-gon).
Cf. A073052, A121598, A272487.

Mathematica
```
First[RealDigits[Csc[Pi/7]/2, 10, 100]]
```

 5/sin(Pi/7) \\ Charles R Greathouse IV, Feb 05 2025

Equals csc(Pi/7)/2 = A121598/2.
Equals 1/(2*sin(Pi/7)) = 1/A272487.
Equals A374971/cos(Pi/7) = A374971/A073052.
Largest of the 6 real-valued roots of 7*x^6-14*x^4+7*x^2-1=0. - R. J. Mathar, Aug 29 2025

A375151 Decimal expansion of the circumradius of a regular 9-gon with unit side length.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Aug 01 2024

			1.46190220008154362611637720668314585193675283...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Regular Polygon.
Wikipedia, Regular polygon.
Index entries for algebraic numbers, degree 6.

Cf. A375152 (apothem), A375153 (sagitta), A256853 (area).
Cf. circumradius of other polygons with unit side length: A020760 (triangle), A010503 (square), A300074 (pentagon), A374957 (heptagon), A285871 (octagon), A001622 (10-gon), A375190 (11-gon), A188887 (12-gon)
Cf. A019879, A121602, A272488.

Mathematica
```
First[RealDigits[Csc[Pi/9]/2, 10, 100]]
```

 5/sin(Pi/9) \\ Charles R Greathouse IV, Feb 05 2025

Equals csc(Pi/9)/2 = A121602/2.
Equals 1/(2*sin(Pi/9)) = 1/A272488.
Equals A375152/cos(Pi/9) = A375152/A019879.
Equals A375152 + A375153.
Largest of the 6 real-valued roots of 3*x^6-9*x^4+6*x^2-1=0. - R. J. Mathar, Aug 29 2025

A375190 Decimal expansion of the circumradius of a regular 11-gon with unit side length.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Aug 04 2024

			1.774732766442111662856831961168975846105376382123...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Regular Polygon.
Wikipedia, Regular polygon.
Index entries for algebraic numbers, degree 10.

Cf. A375191 (apothem), A375192 (sagitta), A256854 (area).
Cf. circumradius of other polygons with unit side length: A020760 (triangle), A010503 (square), A300074 (pentagon), A374957 (heptagon), A285871 (octagon), A375151 (9-gon), A001622 (10-gon), A188887 (12-gon).
Cf. A272489.

First[RealDigits[Csc[Pi/11]/2, 10, 100]]

 5/sin(Pi/11) \\ Charles R Greathouse IV, Feb 05 2025

Equals csc(Pi/11)/2.
Equals 1/(2*sin(Pi/11)) = 1/A272489.
Equals A375191/cos(Pi/11).
Equals A375191 + A375192.

A380736 Decimal expansion of the smallest vertex angle, in radians, in a disdyakis dodecahedron face.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Feb 01 2025

A disdyakis dodecahedron face is a scalene triangle with three acute angles.

			0.6592691539262158395617269590815416456187802510390...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Wikipedia, Disdyakis dodecahedron.

Cf. A380737 (face medium angle), A380738 (face largest angle).

Cf. A010503, A378393, A378712, A378713, A378714, A378715, A380734, A380735.

First[RealDigits[ArcCos[1/12 + Sqrt[2]/2], 10, 100]]

Equals arccos(1/12 + sqrt(2)/2) = arccos(1/12 + A010503).

Equals Pi - A380737 - A380738.

A385506 Decimal expansion of the volume of a triaugmented triangular prism with unit edge.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jul 01 2025

The triaugmented triangular prism is Johnson solid J_51.

			1.140119483078766847782705947481317131020537251141...

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

Cf. A097715 (surface area).

Cf. A002194, A010466, A010503, A120011, A385505.

First[RealDigits[(Sqrt[8] + Sqrt[3])/4, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J51", "Volume"], 10, 100]]

Equals 1/sqrt(2) + sqrt(3)/4 = A010503 + A120011 = (A010466 + A002194)/4.

Equals the largest root of 256*x^4 - 352*x^2 + 25.

A145439 Decimal expansion of Sum_{k>=0} binomial(4k, 2k)/2^(6*k).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Feb 08 2009

			1.11535507165041054076705837455583093794582718446458...

Alexander Apelblat, Tables of Integrals and Series, Harri Deutsch, 1996, 4.1.49.

Cf. A010503, A020760, A020763, A092259.

Maple
```
1/2*(1+1/3*3^(1/2))*2^(1/2);
```

RealDigits[1/Sqrt[2] + 1/Sqrt[6], 10, 120][[1]] (* Amiram Eldar, May 29 2023 *)

1/sqrt(6) + 1/sqrt(2) \\ Michel Marcus, Jan 15 2021

Equals (1+A020760)*A010503.
Equals A020763 + A010503. - Artur Jasinski, Dec 20 2020
The minimal polynomial is 9*x^4 - 12*x^2 + 1. - Joerg Arndt, Sep 20 2023
Equals 2F1(1/4,3/4; 1/2; 1/4). - R. J. Mathar, Aug 02 2024
Equals Product_{k>=1} (1 - (-1)^k/A092259(k)). - Amiram Eldar, Nov 24 2024

Typo in definition corrected by R. J. Mathar, Feb 09 2009

A235362 Decimal expansion of the cube root of 2 divided by 2.

Original entry on oeis.org

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

Offset: 0

Alonso del Arte, Jan 07 2014

Also reciprocal of the real cubic root of 4 and negated real part of either complex cubic root of 2.

			0.6299605249474365823836053...

Richard Zippel, The Weyl Computer Algebra Substrate in Design and Implementation of Symbolic Computation Systems: International Symposium, DISCO '93 Gmunden, Austria, September 15-17, 1993. Proceedings, p. 306.

Cf. A002580, A010503, A020761, A239797.

Digits := 100 ; evalf(1/2^(2/3)) ; # R. J. Mathar, Jan 16 2023

Mathematica
```
RealDigits[1/2^(2/3), 10, 128][[1]]
```

sqrtn(1/4,3) \\ Charles R Greathouse IV, Apr 14 2014

2^(1/3)/2 = 1/2^(2/3) = 1/4^(1/3).
(-2^(1/3)/2 + sqrt(-3)/4^(1/3))^3 = 2.
Equals 1/A005480 = A002580 /2 . - Wolfdieter Lang, Jan 02 2023

A377299 Decimal expansion of the volume of a truncated cube with unit edge length.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Oct 25 2024

			13.599663291074443561074547379645257699991802085...

Eric Weisstein's World of Mathematics, Truncated Cube.
Wikipedia, Truncated cube.

Cf. A377298 (surface area), A294968 (circumradius), A010503 (midradius - 1), A377296 (Dehn invariant, negated).

Cf. A131594.

First[RealDigits[7 + 14*Sqrt[2]/3, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TruncatedCube", "Volume"], 10, 100]]

Equals 7 + (14/3)*sqrt(2) = 7 + 14*A131594.

A023115 Signature sequence of 1/sqrt(2) (arrange the numbers i+j*x (i,j >= 1) in increasing order; the sequence of i's is the signature of x).

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 3, 1, 4, 2, 3, 1, 4, 2, 5, 3, 1, 4, 2, 5, 3, 1, 6, 4, 2, 5, 3, 1, 6, 4, 2, 7, 5, 3, 1, 6, 4, 2, 7, 5, 3, 8, 1, 6, 4, 2, 7, 5, 3, 8, 1, 6, 4, 9, 2, 7, 5, 3, 8, 1, 6, 4, 9, 2, 7, 5, 10, 3, 8, 1, 6, 4, 9, 2, 7, 5, 10, 3, 8, 1, 6, 11, 4, 9, 2, 7, 5, 10, 3, 8

Offset: 1

Clark Kimberling

nonn

C. Kimberling, "Fractal Sequences and Interspersions", Ars Combinatoria, vol. 45 p 157 1997.

T. D. Noe, Table of n, a(n) for n=1..1000
C. Kimberling, Interspersions
Index entries for sequences related to signature sequences

Cf. A010503.

A041084 Numerators of continued fraction convergents to sqrt(50).

Original entry on oeis.org

7, 99, 1393, 19601, 275807, 3880899, 54608393, 768398401, 10812186007, 152139002499, 2140758220993, 30122754096401, 423859315570607, 5964153172084899, 83922003724759193, 1180872205318713601, 16616132878186749607, 233806732499933208099, 3289910387877251662993

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (14,1).

Cf. A010503, A041085.

Numerator[Convergents[Sqrt[50],30]] (* or *) LinearRecurrence[{14,1},{7,99},30] (* Harvey P. Dale, Aug 18 2013 *)
CoefficientList[Series[(7 + x)/(1 - 14 x - x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 29 2013 *)

a(n) = 14*a(n-1)+a(n-2), n>1 ; a(0)=7, a(1)=99 . G.f.: (7+x)/(1-14*x-x^2). - Philippe Deléham, Nov 21 2008

More terms from Colin Barker, Nov 04 2013

cp's OEIS Frontend

A374957 Decimal expansion of the circumradius of a regular heptagon with unit side length.

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A375151 Decimal expansion of the circumradius of a regular 9-gon with unit side length.

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A375190 Decimal expansion of the circumradius of a regular 11-gon with unit side length.

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A380736 Decimal expansion of the smallest vertex angle, in radians, in a disdyakis dodecahedron face.

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A385506 Decimal expansion of the volume of a triaugmented triangular prism with unit edge.

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A145439 Decimal expansion of Sum_{k>=0} binomial(4*k, 2*k)/2^(6*k).

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A235362 Decimal expansion of the cube root of 2 divided by 2.

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Maple

Mathematica

A145439 Decimal expansion of Sum_{k>=0} binomial(4k, 2k)/2^(6*k).