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Egyptian fraction representation of sqrt(53) ( A010506) using a greedy function.
+20
0
7, 4, 34, 1433, 3473810, 16229351336487, 949514635841230182654078450, 2889844410885034994651072554166092838631734010754362047, 90303610423494587890114446343335205731154007285533876023746429382538260256932049359769872513411427600496627202
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[ 53]]
Numerators of continued fraction convergents to sqrt(53).
+10
11
7, 22, 29, 51, 182, 2599, 7979, 10578, 18557, 66249, 946043, 2904378, 3850421, 6754799, 24114818, 344362251, 1057201571, 1401563822, 2458765393, 8777860001, 125348805407, 384824276222, 510173081629, 894997357851, 3195165155182, 45627309530399
COMMENTS
The terms of this sequence can be constructed with the terms of sequence A086902. For the terms of the periodical sequence of the continued fraction for sqrt(53) see A010139. We observe that its period is five. The decimal expansion of sqrt(53) is A010506. - Johannes W. Meijer, Jun 12 2010
LINKS
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,364,0,0,0,0,1).
FORMULA
G.f.: -(x^9-7*x^8+22*x^7-29*x^6+51*x^5+182*x^4+51*x^3+29*x^2+22*x+7) / (x^10+364*x^5-1). - Colin Barker, Sep 26 2013
MATHEMATICA
Numerator[Convergents[Sqrt[53], 30]] (* Harvey P. Dale, Sep 24 2013 *) CoefficientList[Series[-(x^9 - 7 x^8 + 22 x^7 - 29 x^6 + 51 x^5 + 182 x^4 + 51 x^3 + 29 x^2 + 22 x + 7)/(x^10 + 364 x^5 - 1), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 27 2013 *)
Denominators of continued fraction convergents to sqrt(53).
+10
11
1, 3, 4, 7, 25, 357, 1096, 1453, 2549, 9100, 129949, 398947, 528896, 927843, 3312425, 47301793, 145217804, 192519597, 337737401, 1205731800, 17217982601, 52859679603, 70077662204, 122937341807, 438889687625, 6267392968557, 19241068593296, 25508461561853
COMMENTS
The terms of this sequence can be constructed with the terms of sequence A054413. For the terms of the periodic sequence of the continued fraction for sqrt(53) see A010139. We observe that its period is five. The decimal expansion of sqrt(53) is A010506. - Johannes W. Meijer, Jun 12 2010
LINKS
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,364,0,0,0,0,1).
FORMULA
G.f.: -(x^8-3*x^7+4*x^6-7*x^5+25*x^4+7*x^3+4*x^2+3*x+1) / (x^10+364*x^5-1). - Colin Barker, Sep 26 2013
MAPLE
convert(sqrt(53), confrac, 30, cvgts): denom(cvgts); # Wesley Ivan Hurt, Dec 17 2013
MATHEMATICA
LinearRecurrence[{0, 0, 0, 0, 364, 0, 0, 0, 0, 1}, {1, 3, 4, 7, 25, 357, 1096, 1453, 2549, 9100}, 30] (* Harvey P. Dale, Nov 13 2019 *)
Decimal expansion of (7+sqrt(53))/2.
+10
9
7, 1, 4, 0, 0, 5, 4, 9, 4, 4, 6, 4, 0, 2, 5, 9, 1, 3, 5, 5, 4, 8, 6, 5, 1, 2, 4, 5, 7, 6, 3, 5, 1, 6, 3, 9, 6, 8, 8, 8, 8, 3, 4, 8, 4, 1, 2, 8, 8, 2, 3, 8, 7, 1, 9, 1, 8, 9, 0, 9, 0, 8, 9, 5, 6, 4, 2, 0, 5, 7, 8, 6, 9, 3, 1, 2, 4, 5, 2, 5, 9, 1, 6, 6, 4, 7, 8, 9, 7, 0, 4, 5, 4, 0, 4, 6, 3, 3, 7, 6, 0, 9, 6, 3, 1
COMMENTS
Continued fraction expansion of (7+sqrt(53))/2 is A010727. This is the shape of a 7-extension rectangle; see A188640 for definitions. [From Clark Kimberling, Apr 09 2011]
FORMULA
Equals lim_{n->oo} S(n, sqrt(53))/S(n-1, sqrt(53)), with the S-Chebyshev polynomials (see A049310). - Wolfdieter Lang, Nov 15 2023
EXAMPLE
(7+sqrt(53))/2 = 7.14005494464025913554...
MATHEMATICA
r=7; t = (r + (4+r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]
RealDigits[(7+Sqrt[53])/2, 10, 120][[1]] (* Harvey P. Dale, Nov 03 2024 *)
CROSSREFS
Cf. A010506 (decimal expansion of sqrt(53)), A010727 (all 7's sequence).
Continued fraction for sqrt(53).
+10
4
7, 3, 1, 1, 3, 14, 3, 1, 1, 3, 14, 3, 1, 1, 3, 14, 3, 1, 1, 3, 14, 3, 1, 1, 3, 14, 3, 1, 1, 3, 14, 3, 1, 1, 3, 14, 3, 1, 1, 3, 14, 3, 1, 1, 3, 14, 3, 1, 1, 3, 14, 3, 1, 1, 3, 14, 3, 1, 1, 3, 14, 3, 1, 1, 3, 14, 3, 1, 1, 3, 14, 3, 1, 1, 3
EXAMPLE
7.280109889280518271097302491... = 7 + 1/(3 + 1/(1 + 1/(1 + 1/(3 + ...)))). - Harry J. Smith, Jun 06 2009
MATHEMATICA
PadRight[{7}, 120, {14, 3, 1, 1, 3}] (* Harvey P. Dale, May 22 2016 *)
PROG
(PARI) { allocatemem(932245000); default(realprecision, 21000); x=contfrac(sqrt(53)); for (n=0, 20000, write("b010139.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 06 2009
Decimal expansion of sqrt(635918528029).
+10
4
7, 9, 7, 4, 4, 5, 0, 0, 0, 0, 0, 2, 5, 0, 8, 0, 0, 9, 9, 5, 6, 7, 9, 5, 5, 8, 4, 5, 7, 7, 0, 2, 8, 2, 6, 7, 9, 1, 1, 8, 8, 3, 1, 4, 7, 5, 2, 4, 6, 2, 4, 2, 1, 7, 4, 8, 3, 7, 3, 9, 2, 0, 0, 9, 2, 3, 7, 7, 2, 6, 0, 4, 9, 3, 7, 1, 7, 8, 6, 4, 0, 9, 4, 7, 9, 3, 8, 5, 3, 3, 2, 5, 5, 2, 2, 9, 5, 9, 7, 7, 3, 9, 3, 0, 0
COMMENTS
Continued fraction expansion of sqrt(635918528029) is 797445 followed by (repeat 398722, 1, 1, 398722, 1594890).
sqrt(635918528029) = sqrt(17)*sqrt(53)*sqrt(193)*sqrt(3656953).
EXAMPLE
sqrt(635918528029) = 797445.00000250800995679558...
MATHEMATICA
First[RealDigits[Sqrt[635918528029], 10, 120]] (* Paolo Xausa, Jan 09 2024 *)
CROSSREFS
Cf. A010473 (decimal expansion of sqrt(17)), A010506 (decimal expansion of sqrt(53)), A177272 (decimal expansion of sqrt(193)), A177273 (decimal expansion of sqrt(3656953)), A177274 (continued fraction expansion of (684125+sqrt(635918528029))/1033802), A177270 (decimal expansion of (684125+sqrt(635918528029))/1033802).
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