A010525 -id:A010525

A248297

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

+20
0

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

Table of n, a(n) for n=0..8.

MATHEMATICA

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]]

CROSSREFS

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

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

KEYWORD

nonn

AUTHOR

Robert G. Wilson v, Oct 04 2014

STATUS

approved

A351480

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

+10
8

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

2,2

LINKS

Table of n, a(n) for n=2..91.

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

FORMULA

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

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

EXAMPLE

17.2095556595921536435519902312844...

MATHEMATICA

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

CROSSREFS

Cf. A010525, A082640.

Cf. A351481, A351482, A351483.

Cf. A351484, A351485, A351486, A351487, A351488.

KEYWORD

nonn,cons

AUTHOR

Stefano Spezia, Feb 12 2022

STATUS

approved

A351481

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

+10
8

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..90.

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

FORMULA

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

EXAMPLE

2.052568971612735665107871540478655871...

MATHEMATICA

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

PROG

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

CROSSREFS

Cf. A010525, A082640.

Cf. A351480, A351482, A351483.

Cf. A351484, A351485, A351486, A351487, A351488.

KEYWORD

nonn,cons

AUTHOR

Stefano Spezia, Feb 12 2022

STATUS

approved

A010151

Continued fraction for sqrt(73).

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

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).

EXAMPLE

8.544003745317531167871648326... = 8 + 1/(1 + 1/(1 + 1/(5 + 1/(5 + ...)))). - Harry J. Smith, Jun 08 2009

MATHEMATICA

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

PROG

(PARI) { 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

CROSSREFS

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

KEYWORD

nonn,cofr

AUTHOR

N. J. A. Sloane

STATUS

approved

A176980

Decimal expansion of sqrt(365).

+10
3

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

2,2

COMMENTS

Continued fraction expansion of sqrt(365) is A040345.

LINKS

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

EXAMPLE

sqrt(365) = 19.10497317454280017916...

CROSSREFS

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

KEYWORD

cons,nonn

AUTHOR

Klaus Brockhaus, Apr 30 2010

STATUS

approved

A041128

Numerators of continued fraction convergents to sqrt(73).

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

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).

FORMULA

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

MATHEMATICA

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 *)

CROSSREFS

Cf. A010525, A041129.

KEYWORD

nonn,cofr,frac,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Colin Barker, Nov 08 2013

STATUS

approved

A041129

Denominators of continued fraction convergents to sqrt(73).

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

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).

FORMULA

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

MATHEMATICA

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 *)

PROG

(Magma) 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

CROSSREFS

Cf. A041128, A010151, A020830, A010525.

KEYWORD

nonn,frac,easy

AUTHOR

N. J. A. Sloane.

STATUS

approved