A010525 -id:A010525 - cp's OEIS frontend

A248297 Egyptian fraction representation of sqrt(73) (A010525) using a greedy function.

8, 2, 23, 1904, 3644794, 253138275595730, 299921681006149892361129426137, 319157637936684764321170119844052189479588993114762538993037, 104022456806315370788933277888878173955194511356798258776365960524644747879084195850803592853844837028709668458856157018

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[ 73]]

A351480 Decimal expansion of (611 + sqrt(73))/36.

Original entry on oeis.org

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

Offset: 2

Stefano Spezia, Feb 12 2022

			17.2095556595921536435519902312844...

S. Yu. Orevkov, Counting lattice triangulations: Fredholm equations in combinatorics, arXiv:2201.12827 [math.CO], 2022. See Theorem 1, p. 2.
Index entries for algebraic numbers, degree 2

Cf. A010525, A082640.
Cf. A351481, A351482, A351483.
Cf. A351484, A351485, A351486, A351487, A351488.

First[RealDigits[(611+Sqrt[73])/36,10,90]]

Equals lim_{n->oo} A082640(2, n)^(1/n).

Equals 288*x_2, where x_2 is the largest root of 5184*x^2 - 611*x + 18.

A351481 Decimal expansion of log_2((611 + sqrt(73))/36)/2.

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Feb 12 2022

			2.052568971612735665107871540478655871...

S. Yu. Orevkov, Counting lattice triangulations: Fredholm equations in combinatorics, arXiv:2201.12827 [math.CO], 2022. See Theorem 1, p. 2.
Index entries for transcendental numbers

Cf. A010525, A082640.
Cf. A351480, A351482, A351483.
Cf. A351484, A351485, A351486, A351487, A351488.

First[RealDigits[N[Log[2,(611+Sqrt[73])/36]/2,90]]]

log((611 + sqrt(73))/36)/log(4) \\ Charles R Greathouse IV, Oct 31 2023

Equals log_2(alpha)/2, where alpha = lim_{n->oo} A082640(2, n)^(1/n).

A010151 Continued fraction for sqrt(73).

Original entry on oeis.org

8, 1, 1, 5, 5, 1, 1, 16, 1, 1, 5, 5, 1, 1, 16, 1, 1, 5, 5, 1, 1, 16, 1, 1, 5, 5, 1, 1, 16, 1, 1, 5, 5, 1, 1, 16, 1, 1, 5, 5, 1, 1, 16, 1, 1, 5, 5, 1, 1, 16, 1, 1, 5, 5, 1, 1, 16, 1, 1, 5, 5, 1, 1, 16, 1, 1, 5, 5, 1, 1, 16, 1, 1, 5, 5, 1, 1

Offset: 0

N. J. A. Sloane

			8.544003745317531167871648326... = 8 + 1/(1 + 1/(1 + 1/(5 + 1/(5 + ...)))). - _Harry J. Smith_, Jun 08 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, 0, 1).

Cf. A010525 Decimal expansion. - Harry J. Smith, Jun 08 2009

ContinuedFraction[Sqrt[73],300] (* Vladimir Joseph Stephan Orlovsky, Mar 08 2011 *)

{ allocatemem(932245000); default(realprecision, 20000); x=contfrac(sqrt(73)); for (n=0, 20000, write("b010151.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 08 2009

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.

A041128 Numerators of continued fraction convergents to sqrt(73).

Original entry on oeis.org

8, 9, 17, 94, 487, 581, 1068, 17669, 18737, 36406, 200767, 1040241, 1241008, 2281249, 37740992, 40022241, 77763233, 428838406, 2221955263, 2650793669, 4872748932, 80614776581, 85487525513, 166102302094, 915999035983, 4746097482009, 5662096517992

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 (0,0,0,0,0,0,2136,0,0,0,0,0,0,1).

Cf. A010525, A041129.

Numerator[Convergents[Sqrt[73], 30]] (* Vincenzo Librandi, Oct 29 2013 *)
LinearRecurrence[{0,0,0,0,0,0,2136,0,0,0,0,0,0,1},{8,9,17,94,487,581,1068,17669,18737,36406,200767,1040241,1241008,2281249},30] (* Harvey P. Dale, Jul 12 2023 *)

G.f.: -(x^13 -8*x^12 +9*x^11 -17*x^10 +94*x^9 -487*x^8 +581*x^7 +1068*x^6 +581*x^5 +487*x^4 +94*x^3 +17*x^2 +9*x +8) / (x^14 +2136*x^7 -1). - Colin Barker, Nov 08 2013

More terms from Colin Barker, Nov 08 2013

A041129 Denominators of continued fraction convergents to sqrt(73).

Original entry on oeis.org

1, 1, 2, 11, 57, 68, 125, 2068, 2193, 4261, 23498, 121751, 145249, 267000, 4417249, 4684249, 9101498, 50191739, 260060193, 310251932, 570312125, 9435245932, 10005558057, 19440803989, 107209578002, 555488693999, 662698272001, 1218186966000, 20153689728001

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 (0,0,0,0,0,0,2136,0,0,0,0,0,0,1).

Cf. A041128, A010151, A020830, A010525.

I:=[1,1,2,11,57,68,125,2068,2193,4261,23498,121751,145249, 267000]; [n le 14 select I[n] else 2136*Self(n-7)+Self(n-14): n in [1..40]]; // Vincenzo Librandi, Dec 11 2013

Denominator/@Convergents[Sqrt[73], 50] (* Vladimir Joseph Stephan Orlovsky, Jul 05 2011 *)
CoefficientList[Series[-(x^12 - x^11 + 2 x^10 - 11 x^9 + 57 x^8 - 68 x^7 + 125 x^6 + 68 x^5 + 57 x^4 + 11 x^3 + 2 x^2 + x + 1)/(x^14 + 2136 x^7 - 1), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 11 2013 *)

G.f.: -(x^12 -x^11 +2*x^10 -11*x^9 +57*x^8 -68*x^7 +125*x^6 +68*x^5 +57*x^4 +11*x^3 +2*x^2 +x +1) / (x^14 +2136*x^7 -1). - Colin Barker, Nov 13 2013

a(n) = 2136*a(n-7) + a(n-14). - Vincenzo Librandi, Dec 11 2013

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A248297 Egyptian fraction representation of sqrt(73) (A010525) using a greedy function.

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A351480 Decimal expansion of (611 + sqrt(73))/36.

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A351481 Decimal expansion of log_2((611 + sqrt(73))/36)/2.

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A010151 Continued fraction for sqrt(73).

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A176980 Decimal expansion of sqrt(365).

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A041128 Numerators of continued fraction convergents to sqrt(73).

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A041129 Denominators of continued fraction convergents to sqrt(73).

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