A002194 -id:A002194 - cp's OEIS frontend

A386690 Decimal expansion of the surface area of a diminished rhombicosidodecahedron with unit edges.

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

Offset: 2

Paolo Xausa, Jul 29 2025

The diminished rhombicosidodecahedron is Johnson solid J_76.

Also the surface area of a paragyrate diminished rhombicosidodecahedron, a metagyrate diminished rhombicosidodecahedron and a bigyrate diminished rhombicosidodecahedron (Johnson solids J_77, J_78 and J_79, respectively) with unit edges.

			58.11465077780005950750278197201400152953339093...

Paolo Xausa, Table of n, a(n) for n = 2..10000
Wikipedia, Bigyrate diminished rhombicosidodecahedron.
Wikipedia, Diminished rhombicosidodecahedron.
Wikipedia, Metagyrate diminished rhombicosidodecahedron.
Wikipedia, Paragyrate diminished rhombicosidodecahedron.

Cf. A386689 (volume).

Cf. A002194, A010476, A344149, A386692, A386694.

First[RealDigits[25 + (15*Sqrt[3] + 10*Sqrt[#] + 11*Sqrt[5*#])/4 & [5 + Sqrt[20]], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J76", "SurfaceArea"], 10, 100]]

Equals 25 + (15*sqrt(3) + 10*sqrt(5 + 2*sqrt(5)) + 11*sqrt(5*(5 + 2*sqrt(5))))/4 = 25 + (15*A002194 + 10*sqrt(5 + A010476) + 11*sqrt(5*(5 + A010476)))/4.

Equals the largest root of 256*x^8 - 51200*x^7 + 4070400*x^6 - 162560000*x^5 + 3311844000*x^4 - 27184400000*x^3 - 92251037500*x^2 + 2593051875000*x - 8774179671875.

A386692 Decimal expansion of the surface area of a parabidiminished rhombicosidodecahedron with unit edges.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Jul 30 2025

The parabidiminished rhombicosidodecahedron is Johnson solid J_80.

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

			56.9233187106881294742601885078353260314642655523...

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

Cf. A386691 (volume).

Cf. A002194, A010476, A344149, A386690, A386694.

First[RealDigits[5/2*(8 + Sqrt[3] + 2*Sqrt[#] + Sqrt[5*#]) & [5 + Sqrt[20]], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J80", "SurfaceArea"], 10, 100]]

Equals (5/2)*(8 + sqrt(3) + 2*sqrt(5 + 2*sqrt(5)) + sqrt(5*(5 + 2*sqrt(5)))) = (5/2)*(8 + A002194 + 2*sqrt(5 + A010476) + sqrt(5*(5 + A010476))).

Equals the largest root of x^8 - 160*x^7 + 9000*x^6 - 184000*x^5 - 828750*x^4 + 79100000*x^3 - 718984375*x^2 - 3800625000*x + 55781640625.

A386694 Decimal expansion of the surface area of a tridiminished rhombicosidodecahedron with unit edges.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Jul 31 2025

The tridiminished rhombicosidodecahedron is Johnson solid J_83.

			55.731986643576199441017595043656650533395140173888...

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

Cf. A386693 (volume).

Cf. A002194, A010476, A344149, A386690, A386692.

First[RealDigits[(60 + 5*Sqrt[3] + 30*Sqrt[#] + 9*Sqrt[5*#])/4 & [5 + Sqrt[20]], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J83", "SurfaceArea"], 10, 100]]

Equals (60 + 5*sqrt(3) + 30*sqrt(5 + 2*sqrt(5)) + 9*sqrt(5*(5 + 2*sqrt(5))))/4 = (60 + 5*A002194 + 30*sqrt(5 + A010476) + 9*sqrt(5*(5 + A010476)))/4.

Equals the largest root of 256*x^8 - 30720*x^7 + 844800*x^6 + 20736000*x^5 - 1109916000*x^4 + 6460560000*x^3 + 265641862500*x^2 - 4344667875000*x + 19010422828125.

A004548 Expansion of sqrt(3) in base 3.

Original entry on oeis.org

1, 2, 0, 1, 2, 0, 2, 1, 2, 2, 2, 2, 1, 2, 1, 0, 2, 2, 1, 2, 1, 1, 2, 0, 1, 0, 1, 2, 2, 2, 1, 0, 1, 2, 0, 1, 1, 0, 0, 2, 2, 2, 0, 0, 0, 1, 2, 1, 0, 0, 2, 0, 0, 0, 0, 0, 2, 2, 2, 2, 0, 2, 1, 0, 2, 1, 0, 2, 1, 1, 1, 0, 0, 2, 0, 0, 0, 0, 2, 0, 1, 0, 2, 2, 0, 2, 0, 2, 1, 2, 0, 0, 1, 2, 2, 0, 0, 2, 1

Offset: 1

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 1..10000 (terms 1..2000 from Vincenzo Librandi)
Jason Kimberley, Index of expansions of sqrt(d) in base b

Cf. A002194.

Prune(Reverse(IntegerToSequence(Isqrt(3*3^200), 3))); // Vincenzo Librandi, Apr 29 2017

RealDigits[Sqrt[3], 3, 100][[1]] (* Vincenzo Librandi, Apr 29 2017 *)

A067881 Factorial expansion of sqrt(3) = Sum_{n>=1} a(n)/n!.

Original entry on oeis.org

1, 1, 1, 1, 2, 5, 0, 4, 2, 5, 10, 8, 1, 5, 6, 8, 5, 13, 18, 0, 7, 20, 9, 6, 14, 2, 7, 7, 18, 11, 0, 12, 20, 10, 31, 28, 27, 34, 29, 18, 13, 8, 28, 14, 9, 12, 39, 5, 15, 8, 5, 0, 7, 21, 54, 13, 16, 20, 24, 18, 12, 14, 6, 53, 21, 42, 47, 14, 46, 14, 42, 71, 41, 63, 24, 28, 32, 61, 35

Offset: 1

Benoit Cloitre, Mar 10 2002

			sqrt(3) = 1 + 1/2! + 1/3! + 1/4! + 2/5! + 5/6! + 0/7! + 4/8! + 2/9! + ...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for factorial base representation

Cf. A002194 (decimal expansion), A040001 (continued fraction).

Cf. A009949 (sqrt(2)), A068446 (sqrt(5)), A320839 (sqrt(7)).

SetDefaultRealField(RealField(250));  [Floor(Sqrt(3))] cat [Floor(Factorial(n)*Sqrt(3)) - n*Floor(Factorial((n-1))*Sqrt(3)) : n in [2..80]]; // G. C. Greubel, Dec 10 2018

Digits:=200: a:=n->`if`(n=1,floor(sqrt(3)),floor(factorial(n)*sqrt(3))-n*floor(factorial(n-1)*sqrt(3))): seq(a(n),n=1..90); # Muniru A Asiru, Dec 11 2018

With[{b = Sqrt[3]}, Table[If[n == 1, Floor[b], Floor[n!*b] - n*Floor[(n - 1)!*b]], {n, 1, 100}]] (* G. C. Greubel, Dec 10 2018 *)

default(realprecision, 250); {b = sqrt(3); a(n) = if(n==1, floor(b), floor(n!*b) - n*floor((n-1)!*b))};
for(n=1, 80, print1(a(n), ", ")) \\ G. C. Greubel, Dec 10 2018

apply( A067881(n)=if(n>1,sqrt(precision(3., n*log(n/2.5)\2.3+2))*(n-1)!%1*n\1,1), [1..79]) \\ M. F. Hasler, Dec 14 2018

b=sqrt(3);
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)] # G. C. Greubel, Dec 10 2018

a(1) = 1; for n > 1, a(n) = floor(n!*sqrt(3)) - n*floor((n-1)!*sqrt(3)).

A153596 a(n) = ((5 + sqrt(3))^n - (5 - sqrt(3))^n)/(2*sqrt(3)).

Original entry on oeis.org

1, 10, 78, 560, 3884, 26520, 179752, 1214080, 8186256, 55152800, 371430368, 2500942080, 16837952704, 113358801280, 763153053312, 5137636904960, 34587001876736, 232842006858240, 1567506027294208, 10552536122060800, 71040228620135424

Offset: 1

Al Hakanson (hawkuu(AT)gmail.com), Dec 29 2008

nonn

Third binomial transform of A054485. Fifth binomial transform of A162813 preceded by 1.

Lim_{n -> infinity} a(n)/a(n-1) = 5 + sqrt(3) = 6.73205080756887729....

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (10,-22).

Cf. A002194 (decimal expansion of sqrt(3)), A054485, A162813.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-3); S:=[ ((5+r)^n-(5-r)^n)/(2*r): n in [1..25] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Dec 31 2008

I:=[1,10]; [n le 2 select I[n] else 10*Self(n-1)-22*Self(n-2): n in [1..25]]; // Vincenzo Librandi, Aug 23 2016

Table[Simplify[((5+Sqrt[3])^n -(5-Sqrt[3])^n)/(2*Sqrt[3])], {n,1,25}] (* Vladimir Joseph Stephan Orlovsky, Jan 27 2011, modified by G. C. Greubel, Jun 01 2019 *)
LinearRecurrence[{10,-22},{1,10},25] (* G. C. Greubel, Aug 22 2016 *)

my(x='x+O('x^25)); Vec(x/(1-10*x+22*x^2)) \\ G. C. Greubel, Jun 01 2019

[lucas_number1(n,10,22) for n in range(1, 25)] # Zerinvary Lajos, Apr 26 2009

G.f.: x/(1 - 10*x + 22*x^2). - Klaus Brockhaus, Dec 31 2008 [corrected Oct 11 2009]
a(n) = 10*a(n-1) - 22*a(n-2) for n > 1; a(0)=0, a(1)=1. - Philippe Deléham, Jan 01 2009
E.g.f.: sinh(sqrt(3)*x)*exp(5*x)/sqrt(3). - Ilya Gutkovskiy, Aug 23 2016

Extended beyond a(7) by Klaus Brockhaus, Dec 31 2008

Edited by Klaus Brockhaus, Oct 11 2009

A176325 Decimal expansion of (5+3*sqrt(3))/2.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 15 2010

Continued fraction expansion of (5+3*sqrt(3))/2 is A010721.

a(n) = A104956(n) for n > 2.

			5.09807621135331594029...

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

Cf. A002194 (decimal expansion of sqrt(3)), A104956 (decimal expansion of (3*sqrt(3))/2), A010721 (repeat 5, 10).

SetDefaultRealField(RealField(100)); (5+3*Sqrt(3))/2; // G. C. Greubel, Dec 05 2019

evalf( (5+3*sqrt(3))/2, 100); # G. C. Greubel, Dec 05 2019

RealDigits[(5+3Sqrt[3])/2,10,120][[1]] (* Harvey P. Dale, May 20 2011 *)

default(realprecision, 100); (5+3*sqrt(3))/2 \\ G. C. Greubel, Dec 05 2019

numerical_approx((5+3*sqrt(3))/2, digits=100) # G. C. Greubel, Dec 05 2019

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 *)

A178230 Decimal expansion of sqrt(1086).

Original entry on oeis.org

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

Offset: 2

Klaus Brockhaus, May 23 2010

Continued fraction expansion of sqrt(1086) is 32 followed by (repeat 1, 20, 1, 64).

sqrt(1086) = sqrt(2)*sqrt(3)*sqrt(181).

			sqrt(1086) = 32.95451410656816223481...

Cf. A002193 (decimal expansion of sqrt(2)), A002194 (decimal expansion of sqrt(3)), A178231 (decimal expansion of sqrt(181)), A178229 (decimal expansion of (221+11*sqrt(1086))/490).

A186691 Decimal expansion of Gamma(1+sqrt(3)).

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 25 2011

			1.5850036724803687366369911736565893443414147540253..

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A002194, A186631.

Sqrt(3)*Gamma(Sqrt(3)); // G. C. Greubel, Apr 20 2018

Mathematica
```
RealDigits[N[Sqrt[3]!,200]][[1]]
```

gamma(sqrt(3)+1) \\ Charles R Greathouse IV, Feb 27 2011

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A386690 Decimal expansion of the surface area of a diminished rhombicosidodecahedron with unit edges.

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A386692 Decimal expansion of the surface area of a parabidiminished rhombicosidodecahedron with unit edges.

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A386694 Decimal expansion of the surface area of a tridiminished rhombicosidodecahedron with unit edges.

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A004548 Expansion of sqrt(3) in base 3.

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A067881 Factorial expansion of sqrt(3) = Sum_{n>=1} a(n)/n!.

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A153596 a(n) = ((5 + sqrt(3))^n - (5 - sqrt(3))^n)/(2*sqrt(3)).

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A176325 Decimal expansion of (5+3*sqrt(3))/2.

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