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Egyptian fraction representation of sqrt(74) ( A010526) using a greedy function.
+20
0
8, 2, 10, 431, 196796, 42222589233, 4119127882822681368069, 22394712126990929163352329336575823966927304, 810283246500627303789590552867279442902569752132975902553147296681478084954900646327035
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[ 74]]
Continued fraction for sqrt(74).
+10
3
8, 1, 1, 1, 1, 16, 1, 1, 1, 1, 16, 1, 1, 1, 1, 16, 1, 1, 1, 1, 16, 1, 1, 1, 1, 16, 1, 1, 1, 1, 16, 1, 1, 1, 1, 16, 1, 1, 1, 1, 16, 1, 1, 1, 1, 16, 1, 1, 1, 1, 16, 1, 1, 1, 1, 16, 1, 1, 1, 1, 16, 1, 1, 1, 1, 16, 1, 1, 1, 1, 16, 1, 1, 1, 1
FORMULA
Multiplicative with a(5^e) = 16, and a(p^e) = 1 for p != 5.
Dirichlet g.f.: zeta(s) * (1 + 3/5^(s-1)). (End)
EXAMPLE
8.602325267042626771729473535... = 8 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + ...)))). - Harry J. Smith, Jun 09 2009
MATHEMATICA
PadRight[{8}, 120, {16, 1, 1, 1, 1}] (* Harvey P. Dale, Nov 14 2013 *)
PROG
(PARI) { allocatemem(932245000); default(realprecision, 16000); x=contfrac(sqrt(74)); for (n=0, 20000, write("b010152.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 09 2009
Numerators of continued fraction convergents to sqrt(74).
+10
2
8, 9, 17, 26, 43, 714, 757, 1471, 2228, 3699, 61412, 65111, 126523, 191634, 318157, 5282146, 5600303, 10882449, 16482752, 27365201, 454325968, 481691169, 936017137, 1417708306, 2353725443, 39077315394, 41431040837, 80508356231, 121939397068, 202447753299
LINKS
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,86,0,0,0,0,1).
FORMULA
G.f.: -(x^9-8*x^8+9*x^7-17*x^6+26*x^5+43*x^4+26*x^3+17*x^2+9*x+8) / (x^10+86*x^5-1). - Colin Barker, Nov 05 2013
MATHEMATICA
LinearRecurrence[{0, 0, 0, 0, 86, 0, 0, 0, 0, 1}, {8, 9, 17, 26, 43, 714, 757, 1471, 2228, 3699}, 40] (* Harvey P. Dale, Sep 18 2024 *)
Denominators of continued fraction convergents to sqrt(74).
+10
2
1, 1, 2, 3, 5, 83, 88, 171, 259, 430, 7139, 7569, 14708, 22277, 36985, 614037, 651022, 1265059, 1916081, 3181140, 52814321, 55995461, 108809782, 164805243, 273615025, 4542645643, 4816260668, 9358906311, 14175166979, 23534073290, 390720339619, 414254412909
LINKS
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,86,0,0,0,0,1).
FORMULA
G.f.: -(x^4-3*x^3+4*x^2-2*x+1)*(x^4+2*x^3+4*x^2+3*x+1) / (x^10+86*x^5-1). - Colin Barker, Nov 13 2013
MATHEMATICA
CoefficientList[Series[-(x^4 - 3 x^3 + 4 x^2 - 2 x + 1) (x^4 + 2 x^3 + 4 x^2 + 3 x + 1)/(x^10 + 86 x^5 - 1), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 11 2013 *) LinearRecurrence[{0, 0, 0, 0, 86, 0, 0, 0, 0, 1}, {1, 1, 2, 3, 5, 83, 88, 171, 259, 430}, 50] (* Harvey P. Dale, Nov 09 2017 *)
PROG
(Magma) I:=[1, 1, 2, 3, 5, 83, 88, 171, 259, 430]; [n le 10 select I[n] else 86*Self(n-5)+Self(n-10): n in [1..40]]; // Vincenzo Librandi, Dec 11 2013
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