A002163 -id:A002163 - cp's OEIS frontend

A153386 Decimal expansion of Sum_{n>=1} 1/Fibonacci(2*n).

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

Offset: 1

Eric W. Weisstein, Dec 25 2008

			1.535370508836252985029852896651599006367...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.14.1, p. 358.

Richard André-Jeannin, Problem B-745, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 31, No. 3 (1993), p. 277; Fun with Unit Fractions, Solution to Problem B-745 by Paul S. Bruckman, ibid., Vol. 33, No. 1 (1995), p. 86.
A. F. Horadam, Elliptic Functions and Lambert Series in the Summation of Reciprocals in Certain Recurrence-Generated Sequences, The Fibonacci Quarterly, Vol. 26, No. 2 (1988), pp. 98-114.

Cf. A000045, A001906 (Fibonacci(2*n)), A065563.

Cf. A000796, A001113, A002163, A002390, A016628, A079586, A153387, A256178, A265288, A360928.

rd[k_] := rd[k] = RealDigits[ N[ Sum[ 1/Fibonacci[2*n], {n, 1, 2^k}], 105]][[1]]; rd[k = 4]; While[ rd[k] != rd[k - 1], k++]; rd[k] (* Jean-François Alcover, Oct 29 2012 *)
RealDigits[Sqrt[5] * (Log[5] + 2*QPolyGamma[0, 1, 1/GoldenRatio^4] - 4*QPolyGamma[0, 1, 1/GoldenRatio^2]) / (8*ArcCsch[2]), 10, 105][[1]] (* Vaclav Kotesovec, Feb 26 2023 *)

sumpos(n=1, 1/fibonacci(2*n)) \\ Michel Marcus, Sep 04 2021

Equals sqrt(5) * (L((3-sqrt(5))/2) - L((7-3*sqrt(5))/2)), where L(x) = Sum_{k>=1} x^k/(1-x^k) (Horadam, 1988, equation (4.6)). - Amiram Eldar, Oct 04 2020
From Gleb Koloskov, Sep 04 2021: (Start)
Equals 1/2 + (sqrt(5)/log(phi))*(log(5)/8 + 3*Integral_{x=0..infinity} sin(x)/((4*sin(x)^2+5)*(exp(Pi*x/log(phi))-1)) dx), where phi = (1+sqrt(5))/2 = A001622.
Equals 1/2 + (A002163/A002390)*(A016628/8 + 3*Integral_{x=0..infinity} sin(x)/((4*sin(x)^2+5)*(A001113^(A000796*x/A002390)-1)) dx). (End)
Equals 1 + Sum_{n>=1} 1/A065563(2*n-1) (André-Jeannin, 1993). - Amiram Eldar, Jan 15 2022
From Peter Bala, Aug 17 2022: (Start)
Equals 5/3 - 3*Sum_{n >= 1} 1/(F(2*n)*F(2*n+2)*F(2*n+4)), where F(n) = Fibonacci(n).
Conjecture: Equals 151/96 - 6*Sum_{n >= 1} 1/(F(2*n)*F(2*n+4)*F(2*n+6)). (End)
Equals A360928 * sqrt(5). - Kevin Ryde, Feb 27 2023

A134972 Decimal expansion of 2 divided by golden ratio = 2/phi = 4/(1 + sqrt(5)) = 2*(-1 + phi).

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Nov 15 2007

Convergents are 4/2, 8/8, 32/24, 96/80, 320/256, 1024/832, 3328/2688, 10752/8704, 34816/28160, 112640/91136, 364544/294912, 1179648/954368, 3817472/3088384, 12353536/9994240, ... = A209084/A063727. - Seiichi Kirikami, Mar 14 2012

2*(-1 + phi) is an integer in the quadratic number field Q(sqrt(5)). - Wolfdieter Lang, Feb 16 2016

			1.236067977499789696...

Michael Penn, How large is the blue ▲, YouTube video, 2021.
Index entries for algebraic numbers, degree 2.

Cf. A001622, A019863, A063727, A209084, A033887.

RealDigits[ N[4/(1+Sqrt[5]), 150] ] [ [1] ] (* Seiichi Kirikami, Mar 14 2012 *)

4/(1+sqrt(5)) \\ Altug Alkan, Apr 11 2016

Equals A134945 - 2 = A002163 - 1 = A098317 - 3. - R. J. Mathar, Oct 27 2008
2*(-1 + A001622). - Wolfdieter Lang, Feb 17 2016
Equals the harmonic mean of 1 and phi, 2*phi/(1+phi). - Stanislav Sykora, Apr 11 2016
From Christian Katzmann, Mar 19 2018: (Start)
Equals Sum_{n>=0} (15*(2*n)!-8*n!^2)/(n!^2*3^(2*n+2)).
Equals -1 + Sum_{n>=0} 5*(2*n)!/(n!^2*3^(2*n+1)). (End)
Equals 1/A019863. - R. J. Mathar, Jan 17 2021
Equals 2*sin(Pi/5)/sin(2*Pi/5) = hypergeom([1/5, 3/5], [7/5], 1) = hypergeom([-1/5, -3/5], [3/5], 1). - Peter Bala, Mar 04 2022

A290565 Decimal expansion of sum of reciprocal golden rectangle numbers.

Original entry on oeis.org

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

Offset: 1

Bobby Jacobs and Robert G. Wilson v, Aug 06 2017

The constant k in A277266 such that A277266(n) ~ k*n.

			1/(1*1) + 1/(1*2) + 1/(2*3) + 1/(3*5) + ... = 1 + 1/2 + 1/6 + 1/15 + ... = 1.77387758328513234380...

Cf. A000045, A000796, A001622, A001654, A002163, A002390, A079586, A094214, A265290, A277266.

RealDigits[ Sum[1/(Fibonacci[k]*Fibonacci[k + 1]), {k, 265}], 10, 111][[1]]

suminf(n=1, 1/(fibonacci(n)*fibonacci(n+1))) \\ Michel Marcus, Feb 19 2019

Equals Sum_{n>=1} 1/(Fibonacci(n)*Fibonacci(n+1)).
Equals lim_{n->infinity} A277266(n)/n.
Equals 2 * (Sum_{k>=1} 1/(phi^k * F(k))) - 1/phi = 2 * A265290 - A094214, where phi is the golden ratio (A001622) and F(k) is the k-th Fibonacci number (A000045). - Amiram Eldar, Oct 05 2020
Equals 3/2 + 10*c*Integral_{x=0..infinity} f(x) dx, where c = sqrt(5)/log(phi) = A002163/A002390, phi = (1+sqrt(5))/2 = A001622, and f(x) = sin(x)/((exp(Pi*x/(2*log(phi)))-1)*(7-2*cos(x))*(3+2*cos(x))). - Gleb Koloskov, Sep 12 2021

More terms from Alois P. Heinz, Aug 06 2017

A010485 Decimal expansion of square root of 30.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Continued fraction expansion is 5 followed by {2, 10} repeated. - Harry J. Smith, Jun 04 2009

			5.477225575051661134569697828008021339527446949979832542268944....

Harry J. Smith, Table of n, a(n) for n = 1..20000
Index entries for algebraic numbers, degree 2

Cf. A040024, continued fraction. - Harry J. Smith, Jun 04 2009

RealDigits[N[Sqrt[30], 200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 22 2011 *)

default(realprecision, 20080); x=sqrt(30); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010485.txt", n, " ", d));  \\ Harry J. Smith, Jun 04 2009

Equals A010464*A002163 = 1/A020787. - R. J. Mathar, Dec 17 2024

A379708 Decimal expansion of the surface area of a disdyakis triacontahedron with unit shorter edge length.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Dec 31 2024

The disdyakis triacontahedron is the dual polyhedron of the truncated icosidodecahedron (great rhombicosidodecahedron).

			94.234632662193735601503506520549159874997310453708...

Paolo Xausa, Table of n, a(n) for n = 2..10000
Eric Weisstein's World of Mathematics, Disdyakis Triacontahedron.
Wikipedia, Disdyakis triacontahedron.

Cf. A379709 (volume), A379710 (inradius), A379388 (midradius), A379711 (dihedral angle).
Cf. A377796 (surface area of a truncated icosidodecahedron (great rhombicosidodecahedron) with unit edge length).
Cf. A002163.

First[RealDigits[Sqrt[22626/5 + 9738/Sqrt[5]], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["DisdyakisTriacontahedron", "SurfaceArea"], 10, 100]]

Equals sqrt(22626/5 + 9738/sqrt(5)) = sqrt(22626/5 + 9738/A002163).

A379709 Decimal expansion of the volume of a disdyakis triacontahedron with unit shorter edge length.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Dec 31 2024

The disdyakis triacontahedron is the dual polyhedron of the truncated icosidodecahedron (great rhombicosidodecahedron).

			84.1819754400481313518959942929339817444032991207...

Paolo Xausa, Table of n, a(n) for n = 2..10000
Eric Weisstein's World of Mathematics, Disdyakis Triacontahedron.
Wikipedia, Disdyakis triacontahedron.

Cf. A379708 (surface area), A379710 (inradius), A379388 (midradius), A379711 (dihedral angle).
Cf. A377797 (volume of a truncated icosidodecahedron (great rhombicosidodecahedron) with unit edge length).
Cf. A002163.

First[RealDigits[Sqrt[88590 + 39612*Sqrt[5]]/5, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["DisdyakisTriacontahedron", "Volume"], 10, 100]]

Equals sqrt(88590 + 39612*sqrt(5))/5 = sqrt(88590 + 39612*A002163)/5.

A384283 Decimal expansion of the volume of a gyroelongated pentagonal cupola with unit edge.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, May 26 2025

The gyroelongated pentagonal cupola is Johnson solid J_24.

			9.07333319388018799314998398101816272215313393060...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Gyroelongated pentagonal cupola.
Index entries for algebraic numbers, degree 8

Cf. A384284 (surface area).

Cf. A002163, A010532, A179590, A179639, A179641, A384138, A384140, A384144, A384213.

First[RealDigits[(5 + Sqrt[80] + 5*Sqrt[2*(Sqrt[650 + 290*Sqrt[5]] - Sqrt[5] - 1)])/6, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J24", "Volume"], 10, 100]]

(5 + 4*sqrt(5) + 5*sqrt(2*(sqrt(650 + 290*sqrt(5)) - sqrt(5) - 1)))/6 \\ Charles R Greathouse IV, Aug 19 2025

Equals (5 + 4*sqrt(5) + 5*sqrt(2*(sqrt(650 + 290*sqrt(5)) - sqrt(5) - 1)))/6 = (5 + A010532 + 5*sqrt(2*(sqrt(650 + 290*A002163) - A002163 - 1)))/6.

Equals the largest real root of 1679616*x^8 - 11197440*x^7 + 27060480*x^6 + 35769600*x^5 - 4456749600*x^4 - 10714248000*x^3 + 3828402000*x^2 + 13859430000*x + 5340175625.

A019934 Decimal expansion of tangent of 36 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Also the decimal expansion of cotangent of 54 degrees. - Mohammad K. Azarian, Jun 30 2013

A quartic integer. - Charles R Greathouse IV, Aug 27 2017

			0.72654252800536088589546675748061874961609239296520...

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Wikipedia, Exact trigonometric constants.
Index entries for algebraic numbers, degree 4.

Cf. A019934, A019952, A019970, A002163, A063226.

RealDigits[Tan[36 Degree],10,120][[1]] (* Harvey P. Dale, Nov 06 2012 *)

tan(Pi/5) \\ Charles R Greathouse IV, Aug 27 2017

This number is sqrt(5-2*sqrt(5)). This number * A019970 = sqrt(5) = A002163. - R. J. Mathar, Jun 18 2006
The smallest positive solution of cos(4*arctan(x)) = cos(6*arctan(x)). - Thomas Olson, Oct 03 2014
Let r(n) = (n - 1)/(n + 1) if n mod 4 = 1, (n + 1)/(n - 1) otherwise; then this constant (A019934) equals with Product_{n>=0} r(10*n+5) = (2/3) * (8/7) * (12/13) * (18/17) * ... - Dimitris Valianatos, Sep 14 2019
Equals Product_{k>=1} (1 + (-1)^k/A063226(k)). - Amiram Eldar, Nov 23 2024
Equals 1/A019952. - Hugo Pfoertner, Nov 23 2024
tan(Pi/5) = A019845 / A019863. - R. J. Mathar, Aug 31 2025
Smallest positive of the 4 real-valued roots of x^4-10*x^2+5=0. (Other A019970). - R. J. Mathar, Aug 31 2025

A242671 Decimal expansion of k2, a Diophantine approximation constant such that the area of the "critical parallelogram" (in this case a square) is 4*k2.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, May 20 2014

Quoting Steven Finch: "The slopes of the 'critical parallelogram' are (1+sqrt(5))/2 [phi] and (1-sqrt(5))/2 [-1/phi]."
Essentially the same as A229780, A134972, A134945, A098317 and A002163. - R. J. Mathar, May 23 2014
Let W_n be the collection of all binary words of length n that do not contain two consecutive 0's. Let r_n be the ratio of the total number of 1's in W_n divided by the total number of letters in W_n. Then lim_{n->oo} r_n = 0.723606... Equivalently, lim_{n->oo} A004798(n)/(n*A000045(n+2)) = 0.723606... - Geoffrey Critzer, Feb 04 2022
The limiting frequency of the digit 0 in the base phi representation of real numbers in the range [0,1], where phi is the golden ratio (A001622) (Rényi, 1957). - Amiram Eldar, Mar 18 2025

			k2 = 0.723606797749978969640917366873127623544...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.23, p. 176.

Alfréd Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar., Vol.8, No. 3-4 (1957), pp. 477-493.
Index entries for algebraic numbers, degree 2.

Cf. A000032, A000045, A001622, A002163, A004798, A020762, A081007, A094214, A094874, A098317, A134972, A229780, A244847, A296184, A300074, A344212.

RealDigits[(1+1/Sqrt[5])/2, 10, 100] // First

(1 + 1/sqrt(5))/2 \\ Stefano Spezia, Dec 07 2024

Equals (1 + 1/sqrt(5))/2.
Equals 1/A094874. - Michel Marcus, Dec 01 2018
From Amiram Eldar, Feb 11 2022: (Start)
Equals phi/sqrt(5), where phi is the golden ratio (A001622).
Equals lim_{k->oo} Fibonacci(k+1)/Lucas(k). (End)
From Amiram Eldar, Nov 28 2024: (Start)
Equals A344212/2 = A296184/5 = A300074^2 = sqrt(A229780).
Equals Product_{k>=1} (1 - 1/A081007(k)). (End)
Equals 1 - A244847. - Amiram Eldar, Mar 18 2025

A378973 Decimal expansion of the surface area of a triakis icosahedron with unit shorter edge length.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Dec 13 2024

The triakis icosahedron is the dual polyhedron of the truncated dodecahedron.

			26.228595976743751681458195104356801731865266699519...

Paolo Xausa, Table of n, a(n) for n = 2..10000
Eric Weisstein's World of Mathematics, Triakis Icosahedron.
Wikipedia, Triakis icosahedron.

Cf. A378974 (volume), A378975 (inradius), A378976 (midradius), A378977 (dihedral angle).
Cf. A377694 (surface area of a truncated dodecahedron with unit edge length).
Cf. A002163.

First[RealDigits[3*Sqrt[(173 - 9*Sqrt[5])/2], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TriakisIcosahedron", "SurfaceArea"], 10, 100]]

Equals 3*sqrt((173 - 9*sqrt(5))/2) = 3*sqrt((173 - 9*A002163)/2).

cp's OEIS Frontend

A153386 Decimal expansion of Sum_{n>=1} 1/Fibonacci(2*n).

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A134972 Decimal expansion of 2 divided by golden ratio = 2/phi = 4/(1 + sqrt(5)) = 2*(-1 + phi).

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A290565 Decimal expansion of sum of reciprocal golden rectangle numbers.

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A010485 Decimal expansion of square root of 30.

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A379708 Decimal expansion of the surface area of a disdyakis triacontahedron with unit shorter edge length.

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A379709 Decimal expansion of the volume of a disdyakis triacontahedron with unit shorter edge length.

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A384283 Decimal expansion of the volume of a gyroelongated pentagonal cupola with unit edge.

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