A010497 -id:A010497 - cp's OEIS frontend

A248269 Egyptian fraction representation of sqrt(43) (A010497) using a greedy function.

6, 2, 18, 532, 305858, 137859230710, 22012211318177566410441, 1147928569154887244380386940705198857524244457, 54505440157936785019731226309482186897275025107764309863984976644953861019275801793173245974

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[ 43]]

A010134 Continued fraction for sqrt(43).

Original entry on oeis.org

6, 1, 1, 3, 1, 5, 1, 3, 1, 1, 12, 1, 1, 3, 1, 5, 1, 3, 1, 1, 12, 1, 1, 3, 1, 5, 1, 3, 1, 1, 12, 1, 1, 3, 1, 5, 1, 3, 1, 1, 12, 1, 1, 3, 1, 5, 1, 3, 1, 1, 12, 1, 1, 3, 1, 5, 1, 3, 1, 1, 12, 1, 1, 3, 1, 5, 1, 3, 1, 1, 12, 1, 1, 3, 1, 5, 1, 3

Offset: 0

N. J. A. Sloane

			6.557438524302000652344109997... = 6 + 1/(1 + 1/(1 + 1/(3 + 1/(1 + ...)))) - _Harry J. Smith_, Jun 05 2009

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 276.

Harry J. Smith, Table of n, a(n) for n = 0..20000
Les Tablettes du Chercheur, Problem 364, pp. 11, Mai 15 1891
G. Xiao, Contfrac
Index entries for continued fractions for constants
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,1).

Cf. A010497 (decimal expansion).

ContinuedFraction[Sqrt[43],300] (* Vladimir Joseph Stephan Orlovsky, Mar 06 2011 *)

{ allocatemem(932245000); default(realprecision, 16000); x=contfrac(sqrt(43)); for (n=0, 20000, write("b010134.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 05 2009

Vec((6*x^10+x^9+x^8+3*x^7+x^6+5*x^5+x^4+3*x^3+x^2+x+6)/(-x^10+1) + O(x^100)) \\ Colin Barker, Nov 01 2013

From Colin Barker, Nov 01 2013: (Start)
G.f.: (6*x^10+x^9+x^8+3*x^7+x^6+5*x^5+x^4+3*x^3+x^2+x+6)/(1-x^10).
a(n) = a(n-10) for n>10. (End)

A228932 Optimal ascending continued fraction expansion of sqrt(43) - 6.

Original entry on oeis.org

2, 9, 30, 60, 122, -878, 11429, 35241, -177141, 709582, -3123032, -1157723745, 3237738813, -16178936725, 33395053634, -71863018424, -153349368674, -386763022623, -8021033029400, 16314606875900, 52522689388692

Offset: 1

Giovanni Artico, Sep 10 2013

See A228929 for the definition of "optimal ascending continued fraction".
In A228931 it is shown that many numbers of the type sqrt(x) seem to present in their expansion a recurrence relation a(n) = a(n-1)^2 - 2 between the terms, starting from some point onward; 43 is the first natural number whose terms don't respect this relation.
The numbers in range 1 .. 200 that exhibit this behavior are 43, 44, 46, 53, 58, 61, 67, 73, 76, 85, 86, 89, 91, 94, 97, 103, 106, 108, 109, 113, 115, 116, 118, 125, 127, 129, 131, 134, 137, 139, 149, 151, 153, 154, 157, 159, 160, 161, 163, 166, 172, 173, 176, 177, 179, 181, 184, 186, 190, 191, 193, 199.
Nevertheless, the expansions of 3*sqrt(43), 9*sqrt(43), and sqrt(43)/5 satisfy the recurrence relation.

			sqrt(43) = 6 + 1/2*(1 + 1/9*(1 + 1/30*(1 + 1/60*(1 + 1/122*(1 - 1/878*(1 + ...)))))).

See A228931.

G. C. Greubel, Table of n, a(n) for n = 1..500

Cf. A010134, A010497, A228929, A228931.

ArticoExp := proc (n, q::posint)::list; local L, i, z; Digits := 50000; L := []; z := frac(evalf(n)); for i to q+1 do if z = 0 then break end if; L := [op(L), round(1/abs(z))*sign(z)]; z := abs(z)*round(1/abs(z))-1 end do; return L end proc
# List the first 8 terms of the expansion of sqrt(43)-6
ArticoExp(sqrt(43),20)

ArticoExp[x_, n_] := Round[1/#] & /@ NestList[Round[1/Abs[#]]*Abs[#] - 1 &, FractionalPart[x], n]; Block[{$MaxExtraPrecision = 50000},
ArticoExp[Sqrt[43] - 6, 20]] (* G. C. Greubel, Dec 26 2016 *)

A041072 Numerators of continued fraction convergents to sqrt(43).

Original entry on oeis.org

6, 7, 13, 46, 59, 341, 400, 1541, 1941, 3482, 43725, 47207, 90932, 320003, 410935, 2374678, 2785613, 10731517, 13517130, 24248647, 304500894, 328749541, 633250435, 2228500846, 2861751281, 16537257251

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Les Tablettes du Chercheur, Problem 364, pp. 11, Mai 15 1891
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,6964,0,0,0,0,0,0,0,0,0,-1).

Cf. A041073, A010497, A010134.

Numerator[Convergents[Sqrt[43], 30]] (* Vincenzo Librandi, Oct 29 2013 *)

G.f.: -(x^19 -6*x^18 +7*x^17 -13*x^16 +46*x^15 -59*x^14 +341*x^13 -400*x^12 +1541*x^11 -1941*x^10 -3482*x^9 -1941*x^8 -1541*x^7 -400*x^6 -341*x^5 -59*x^4 -46*x^3 -13*x^2 -7*x -6) / (x^20 -6964*x^10 +1). - Colin Barker, Nov 04 2013

A041073 Denominators of continued fraction convergents to sqrt(43).

Original entry on oeis.org

1, 1, 2, 7, 9, 52, 61, 235, 296, 531, 6668, 7199, 13867, 48800, 62667, 362135, 424802, 1636541, 2061343, 3697884, 46435951, 50133835, 96569786, 339843193, 436412979, 2521908088, 2958321067, 11396871289, 14355192356, 25752063645, 323379956096, 349132019741

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Les Tablettes du Chercheur, Problem 364, pp. 11, Mai 15 1891.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,6964,0,0,0,0,0,0,0,0,0,-1).

Cf. A010497, A041072.

I:=[1, 1, 2, 7, 9, 52, 61, 235, 296, 531, 6668, 7199, 13867, 48800, 62667, 362135, 424802, 1636541, 2061343, 3697884]; [n le 20 select I[n] else 6964*Self(n-10)-Self(n-20): n in [1..40]]; // Vincenzo Librandi, Dec 10 2013

Denominator[Convergents[Sqrt[43], 40]] (* Vincenzo Librandi, Dec 10 2013 *)

G.f.: -(x^18 -x^17 +2*x^16 -7*x^15 +9*x^14 -52*x^13 +61*x^12 -235*x^11 +296*x^10 -531*x^9 -296*x^8 -235*x^7 -61*x^6 -52*x^5 -9*x^4 -7*x^3 -2*x^2 -x -1) / (x^20 -6964*x^10 +1). - Colin Barker, Nov 12 2013

a(n) = 6964*a(n-10) - a(n-20). - Vincenzo Librandi, Dec 10 2013

More terms from Colin Barker, Nov 12 2013

A177159 Decimal expansion of sqrt(4171).

Original entry on oeis.org

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

Offset: 2

Klaus Brockhaus, May 03 2010

Continued fraction expansion of sqrt(4171) is 64 followed by (repeat 1, 1, 2, 1, 1, 128).

sqrt(4171) = sqrt(43)*sqrt(97).

			sqrt(4171) = 64.58327956987009497650...

Cf. A010497 (decimal expansion of sqrt(43)), A010548 (decimal expansion of sqrt(97)), A177158 (decimal expansion of (103+2*sqrt(4171))/162).

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A248269 Egyptian fraction representation of sqrt(43) (A010497) using a greedy function.

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A010134 Continued fraction for sqrt(43).

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A228932 Optimal ascending continued fraction expansion of sqrt(43) - 6.

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A041072 Numerators of continued fraction convergents to sqrt(43).

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A041073 Denominators of continued fraction convergents to sqrt(43).

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A177159 Decimal expansion of sqrt(4171).

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