A010507 -id:A010507 - cp's OEIS frontend

A248279 Egyptian fraction representation of sqrt(54) (A010507) using a greedy function.

7, 3, 67, 4751, 25076431, 1253373011645810, 9187269148593176940086772749458, 498651977464932900685397060435928260390239175775532045711576034, 321776209073611476881274134051635561805771857820185011672099181310492331070886792488196910194328794077954530415887963244506932

Offset: 0

Robert G. Wilson v, Oct 04 2014

nonn

Egyptian fraction representations of the square roots: A006487, A224231, A248235-A248322.

Egyptian fraction representations of the cube roots: A129702, A132480-A132574.

Egyptian[nbr_] := Block[{lst = {IntegerPart[nbr]}, cons = N[ FractionalPart[ nbr], 2^20], denom, iter = 8}, While[ iter > 0, denom = Ceiling[ 1/cons]; AppendTo[ lst, denom]; cons -= 1/denom; iter--]; lst]; Egyptian[ Sqrt[ 54]]

A010140 Continued fraction for sqrt(54).

Original entry on oeis.org

7, 2, 1, 6, 1, 2, 14, 2, 1, 6, 1, 2, 14, 2, 1, 6, 1, 2, 14, 2, 1, 6, 1, 2, 14, 2, 1, 6, 1, 2, 14, 2, 1, 6, 1, 2, 14, 2, 1, 6, 1, 2, 14, 2, 1, 6, 1, 2, 14, 2, 1, 6, 1, 2, 14, 2, 1, 6, 1, 2, 14, 2, 1, 6, 1, 2, 14, 2, 1, 6, 1, 2, 14, 2, 1

Offset: 0

N. J. A. Sloane

			7.348469228349534294591852224... = 7 + 1/(2 + 1/(1 + 1/(6 + 1/(1 + ...)))). - _Harry J. Smith_, Jun 06 2009

Harry J. Smith, Table of n, a(n) for n = 0..20000
G. Xiao, Contfrac
Index entries for continued fractions for constants
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 1).

Cf. A010507 Decimal expansion. - Harry J. Smith, Jun 06 2009

ContinuedFraction[Sqrt[54],300] (* Vladimir Joseph Stephan Orlovsky, Mar 07 2011 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(sqrt(54)); for (n=0, 20000, write("b010140.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 06 2009

A386739 Decimal expansion of the volume of a sphenocorona with unit edges.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Aug 01 2025

The sphenocorona is Johnson solid J_86.

			1.5153516399764065597284793124718129048228695068...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Sphenocorona.

Cf. A010482 (surface area - 2), A178809 (surface area + 4).

Cf. A010507, A020775, A115754, A386740, A386741.

First[RealDigits[Sqrt[1 + 3*Sqrt[3/2] + Sqrt[13 + Sqrt[54]]]/2, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J86", "Volume"], 10, 100]]

Equals sqrt(1 + 3*sqrt(3/2) + sqrt(13 + 3*sqrt(6)))/2 = sqrt(1 + 3*A115754  + sqrt(13 + A010507))/2.
Equals A386740 - A020775.
Equals the largest real root of 1024*x^8 - 1024*x^6 - 3008*x^4 - 96*x^2 + 9.

A386740 Decimal expansion of the volume of an augmented sphenocorona with unit edges.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Aug 01 2025

The augmented sphenocorona is Johnson solid J_87.

			1.7510539003719224011954274331734292512511481527...

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

Cf. A010502 (surface area - 1).

Cf. A010507, A020775, A115754, A386739, A386741.

First[RealDigits[Sqrt[1 + 3*Sqrt[3/2] + Sqrt[13 + Sqrt[54]]]/2 + 1/Sqrt[18], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J87", "Volume"], 10, 100]]

Equals sqrt(1 + 3*sqrt(3/2) + sqrt(13 + 3*sqrt(6)))/2 + 1/(3*sqrt(2)) = sqrt(1 + 3*A115754  + sqrt(13 + A010507))/2 + A020775.
Equals A386739 + A020775.
Equals the largest real root of 45137758519296*x^16 - 110336743047168*x^14 - 191069246324736*x^12 + 209269081571328*x^10 + 364547659290624*x^8 - 58793017190400*x^6 + 3306865979520*x^4 - 1275399855936*x^2 + 1439671249.

A041092 Numerators of continued fraction convergents to sqrt(54).

Original entry on oeis.org

7, 15, 22, 147, 169, 485, 6959, 14403, 21362, 142575, 163937, 470449, 6750223, 13970895, 20721118, 138297603, 159018721, 456335045, 6547709351, 13551753747, 20099463098, 134148532335, 154247995433

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,970,0,0,0,0,0,-1).

Cf. A010507, A041093.

Numerator[Convergents[Sqrt[54],30]] (* Harvey P. Dale, Jul 18 2013 *)
CoefficientList[Series[- (x^11 - 7 x^10 + 15 x^9 - 22 x^8 + 147 x^7 - 169 x^6 - 485 x^5 - 169 x^4 - 147 x^3 - 22 x^2 - 15 x - 7)/((x^4 - 10 x^2 + 1) (x^8 + 10 x^6 + 99 x^4 + 10 x^2 + 1)), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 25 2013 *)

a(n) = 970*a(n-6)-a(n-12). G.f.: -(x^11-7*x^10+15*x^9-22*x^8+147*x^7-169*x^6-485*x^5-169*x^4-147*x^3-22*x^2-15*x-7)/((x^4-10*x^2+1)*(x^8+10*x^6+99*x^4+10*x^2+1)). [Colin Barker, Jul 18 2012]

A041093 Denominators of continued fraction convergents to sqrt(54).

Original entry on oeis.org

1, 2, 3, 20, 23, 66, 947, 1960, 2907, 19402, 22309, 64020, 918589, 1901198, 2819787, 18819920, 21639707, 62099334, 891030383, 1844160100, 2735190483, 18255302998, 20990493481, 60236289960, 864298552921, 1788833395802, 2653131948723, 17707625088140

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,970,0,0,0,0,0,-1).

Cf. A010507, A041092.

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[54], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Jun 23 2011 *)
Denominator[Convergents[Sqrt[54], 30]] (* Vincenzo Librandi, Oct 24 2013 *)

a(n) = 970*a(n-6)-a(n-12). G.f.: -(x^10-2*x^9+3*x^8-20*x^7+23*x^6-66*x^5-23*x^4-20*x^3-3*x^2-2*x-1)/((x^4-10*x^2+1)*(x^8+10*x^6+99*x^4+10*x^2+1)). [Colin Barker, Jul 18 2012]

A379531 Decimal expansion of (3sqrt(6) - 7)Pi/3.

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Dec 24 2024

Lower bound to the volume of the Meisser's tetrahedral analog of the Reuleaux triangle.

			0.3649161225950003501847169303738650723435020735...

G. D. Chakerian and H. Groemer, Convex bodies of constant width, Convexity and Its Applications, ed. P. M. Gruber and J. M. Wills, Birkhäuser, 1983, pp. 49-96; MR85f:52001.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.10, p. 514.

G. D. Chakerian, Sets of constant width, Pacific J. Math. 19 (1966) 13-21; MR 34 #4986. See p. 14.
Michael I. Shamos, A catalog of the real numbers, (2007). See p. 230.

Cf. A000796, A010464, A010507, A019670.

RealDigits[(3Sqrt[6]-7)Pi/3,10,100][[1]]

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A248279 Egyptian fraction representation of sqrt(54) (A010507) using a greedy function.

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A010140 Continued fraction for sqrt(54).

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A386739 Decimal expansion of the volume of a sphenocorona with unit edges.

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A386740 Decimal expansion of the volume of an augmented sphenocorona with unit edges.

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A041092 Numerators of continued fraction convergents to sqrt(54).

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A041093 Denominators of continued fraction convergents to sqrt(54).

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A379531 Decimal expansion of (3*sqrt(6) - 7)*Pi/3.

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A379531 Decimal expansion of (3sqrt(6) - 7)Pi/3.