A002163 -id:A002163 - cp's OEIS frontend

A386691 Decimal expansion of the volume of a parabidiminished rhombicosidodecahedron with unit edges.

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

Offset: 2

Paolo Xausa, Jul 30 2025

The parabidiminished rhombicosidodecahedron is Johnson solid J_80.

Also the volume of a metabidiminished rhombicosidodecahedron and a gyrate bidiminished rhombicosidodecahedron (Johnson solids J_81 and J_82, respectively) with unit edges.

			36.967233145831580803409780572760635295338486330...

Paolo Xausa, Table of n, a(n) for n = 2..10000
Wikipedia, Gyrate bidiminished rhombicosidodecahedron.
Wikipedia, Metabidiminished rhombicosidodecahedron.
Wikipedia, Parabidiminished rhombicosidodecahedron.

Cf. A386692 (surface area).

Cf. A001622, A002163, A134946, A179590, A185093, A386689, A386693.

First[RealDigits[5/3*(11 + 5*Sqrt[5]), 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J80", "Volume"], 10, 100]]

Equals (5/3)*(11 + 5*sqrt(5)) = (5/3)*(11 + 5*A002163).
Equals A185093 - 2*A179590.
Equals (50/3)*A001622 + 10 = A134946*100 + 10.
Equals the largest root of 9*x^2 - 330*x - 100.

A386693 Decimal expansion of the volume of a tridiminished rhombicosidodecahedron with unit edges.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Jul 31 2025

The tridiminished rhombicosidodecahedron is Johnson solid J_83.

			34.64318782749838767247033146027311780504474075702...

Paolo Xausa, Table of n, a(n) for n = 2..10000
Wikipedia, Tridiminished rhombicosidodecahedron.

Cf. A386694 (surface area).

Cf. A002163, A179590, A185093, A386689, A386691.

First[RealDigits[35/2 + 23/3*Sqrt[5], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J83", "Volume"], 10, 100]]

Equals 35/2 + (23/3)*sqrt(5) = 35/2 + (23/3)*A002163.
Equals A185093 - 3*A179590.
Equals the largest root of 36*x^2 - 1260*x + 445.

A004559 Expansion of sqrt(5) in base 6.

Original entry on oeis.org

2, 1, 2, 2, 5, 5, 3, 5, 5, 3, 1, 5, 1, 3, 0, 3, 3, 4, 3, 1, 2, 4, 5, 1, 4, 3, 2, 0, 3, 4, 0, 2, 4, 0, 1, 3, 4, 5, 4, 0, 2, 5, 2, 1, 3, 2, 2, 3, 2, 0, 3, 3, 2, 5, 0, 2, 1, 5, 4, 4, 1, 1, 0, 1, 3, 2, 1, 5, 5, 0, 1, 0, 0, 0, 4, 5, 3, 1, 4, 1, 1, 2, 5, 1, 4, 2, 5, 0, 0, 0, 0, 1, 1, 3, 4, 5, 1, 3, 5

Offset: 1

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Jason Kimberley, Index of expansions of sqrt(d) in base b

Cf. A002163, A004555, A004556, A004557, A004558.

Cf. A004540, A004549, A004575.

Prune(Reverse(IntegerToSequence(Isqrt(5*6^200), 6))); // Vincenzo Librandi, Jan 08 2018

RealDigits[Sqrt[5],6,120][[1]] (* Harvey P. Dale, Mar 24 2012 *)

Updated by Alois P. Heinz at the suggestion of Kevin Ryde, Feb 19 2012

A023117 Signature sequence of sqrt(5) (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, 2, 3, 1, 4, 2, 5, 3, 1, 6, 4, 2, 7, 5, 3, 1, 8, 6, 4, 2, 9, 7, 5, 3, 1, 10, 8, 6, 4, 2, 11, 9, 7, 5, 3, 12, 1, 10, 8, 6, 4, 13, 2, 11, 9, 7, 5, 14, 3, 12, 1, 10, 8, 6, 15, 4, 13, 2, 11, 9, 7, 16, 5, 14, 3, 12, 1, 10, 8, 17, 6, 15, 4, 13, 2, 11, 9, 18, 7, 16, 5, 14, 3, 12, 1, 10

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. A002163.

A145434 Decimal expansion of Sum_{n>=1} (-1)^(n-1)*n^2/binomial(2n,n).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Feb 08 2009

The numerator in the Apelblat table lacks the square (typo).

			0.125567284722879676...

Alexander Apelblat, Tables of Integrals and Series, Harri Deutsch, (1996), 4.1.40.

Steven Finch, Central Binomial Coefficients [Cached copy, with permission of the author]

Cf. A002163, A002390, A145433.

evalf( 4/25-4/125*5^(1/2)*log(1/2+1/2*5^(1/2)), 120) ;

RealDigits[HypergeometricPFQ[{2, 2, 2}, {1, 3/2}, -1/4]/2, 10, 93] // First
(* or *) RealDigits[4/25 - 4*Sqrt[5]*Log[GoldenRatio]/125, 10, 93] // First (* Jean-François Alcover, Feb 13 2013, updated Oct 27 2014 *)

Equals 4*(5-A002163*A002390)/125.

A176324 Decimal expansion of (15+7*sqrt(5))/6.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 15 2010

Continued fraction expansion of (15+7*sqrt(5))/6 is A010720.

Volume of a triangular hebesphenorotunda (Johnson solid J_92) with unit edges. - Paolo Xausa, Aug 02 2025

			5.10874597374975464581...

G. C. Greubel, Table of n, a(n) for n = 1..1000
Wikipedia, Triangular hebesphenorotunda.

Cf. A002163 (decimal expansion of sqrt(5)), A010720 (repeat 5, 9).

SetDefaultRealField(RealField(100)); (15+7*Sqrt(5))/6; // G. C. Greubel, Dec 05 2019

evalf( (15+7*sqrt(5))/6, 100); # G. C. Greubel, Dec 05 2019

RealDigits[(15+7Sqrt[5])/6,10,120][[1]]  (* Harvey P. Dale, Apr 18 2011 *)

default(realprecision, 100); (15+7*sqrt(5))/6 \\ G. C. Greubel, Dec 05 2019

numerical_approx((15+7*sqrt(5))/6, digits=100) # G. C. Greubel, Dec 05 2019

A176453 Decimal expansion of 4+2*sqrt(5).

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 20 2010

Continued fraction expansion of 4+2*sqrt(5) is A010698 preceded by 8.
a(n) = A010476(n) = A020762(n-1) = A134974(n) for n > 1.
Rajan (2010) claims the variance of a discrete distribution generated by the linear convolution of Fibonacci sequence with itself, saturates to a constant of value 8.4721359. [From Jonathan Vos Post, May 10 2010]

			4+2*sqrt(5) = 8.47213595499957939281...

Arulalan Rajan, Jamadagni, Vittal Rao, Ashok Rao, Convolutions Induced Discrete Probability Distributions and a New Fibonacci Constant, May 6, 2010. [From _Jonathan Vos Post_, May 10 2010]

Cf. A002163 (decimal expansion of sqrt(5)), A010476 (decimal expansion of sqrt(20)), A020762 (decimal expansion of 1/sqrt(5)), A134974 (decimal expansion of 8/(1+sqrt(5))), A010698 (repeat 2, 8).

RealDigits[4+2Sqrt[5],10,120][[1]] (* Harvey P. Dale, Sep 08 2018 *)

A176461 Decimal expansion of sqrt(105).

Original entry on oeis.org

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

Offset: 2

Klaus Brockhaus, Apr 20 2010

Continued fraction expansion of sqrt(105) is A040094.

			sqrt(105) = 10.24695076595959838322...

Cf. A002194, A002163, A010465, A176460, A040094.

RealDigits[Sqrt[105],10,120][[1]] (* Harvey P. Dale, Aug 03 2016 *)

A176524 Decimal expansion of sqrt(235).

Original entry on oeis.org

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

Offset: 2

Klaus Brockhaus, Apr 23 2010

Continued fraction expansion of sqrt(235) is A040219.

			sqrt(235) = 15.32970971675589165655...

Cf. A002163 (decimal expansion of sqrt(5)), A010501 (decimal expansion of sqrt(47)), A176523 (decimal expansion of (45+3*sqrt(235))/10), A040219 (15 followed by (repeat 3, 30)).

RealDigits[Sqrt[235],10,120][[1]] (* Harvey P. Dale, Mar 12 2015 *)

A176980 Decimal expansion of sqrt(365).

Original entry on oeis.org

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

Offset: 2

Klaus Brockhaus, Apr 30 2010

Continued fraction expansion of sqrt(365) is A040345.

			sqrt(365) = 19.10497317454280017916...

Daniel Starodubtsev, Table of n, a(n) for n = 2..10000

Cf. A002163 (decimal expansion of sqrt(5)), A010525 (decimal expansion of sqrt(73)), A176979 (decimal expansion of (15+sqrt(365))/10), A040345.

cp's OEIS Frontend

A386691 Decimal expansion of the volume of a parabidiminished rhombicosidodecahedron with unit edges.

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A386693 Decimal expansion of the volume of a tridiminished rhombicosidodecahedron with unit edges.

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A004559 Expansion of sqrt(5) in base 6.

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A023117 Signature sequence of sqrt(5) (arrange the numbers i+j*x (i,j >= 1) in increasing order; the sequence of i's is the signature of x).

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A145434 Decimal expansion of Sum_{n>=1} (-1)^(n-1)*n^2/binomial(2n,n).

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A176324 Decimal expansion of (15+7*sqrt(5))/6.

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A176453 Decimal expansion of 4+2*sqrt(5).

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A176461 Decimal expansion of sqrt(105).

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