A010533 -id:A010533 - cp's OEIS frontend

A248305 Egyptian fraction representation of sqrt(82) (A010533) using a greedy function.

9, 19, 364, 158568, 7483072370239, 800584801436242461055205607, 804967345737393522659886914511772380605508613608740482, 1952430246641956813527923846249169608538413464343857806735578675242145974375232933703999085491264008473613681

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[ 82]]

A041145 Denominators of continued fraction convergents to sqrt(82).

Original entry on oeis.org

1, 18, 325, 5868, 105949, 1912950, 34539049, 623615832, 11259624025, 203296848282, 3670602893101, 66274148924100, 1196605283526901, 21605169252408318, 390089651826876625, 7043218902136187568, 127168029890278252849, 2296067756927144738850, 41456387654578883552149

Offset: 0

N. J. A. Sloane

For n >= 2, a(n) equals the permanent of the (n-1) X (n-1) tridiagonal matrix with 18's along the main diagonal, and 1's along the superdiagonal and the subdiagonal. - John M. Campbell, Jul 08 2011
a(n) equals the number of words of length n on alphabet {0,1,...,18} avoiding runs of zeros of odd lengths. - Milan Janjic, Jan 28 2015
From Michael A. Allen, May 03 2023: (Start)
Also called the 18-metallonacci sequence; the g.f. 1/(1-k*x-x^2) gives the k-metallonacci sequence.
a(n) is the number of tilings of an n-board (a board with dimensions n X 1) using unit squares and dominoes (with dimensions 2 X 1) if there are 18 kinds of squares available. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Michael A. Allen and Kenneth Edwards, Fence tiling derived identities involving the metallonacci numbers squared or cubed, Fib. Q. 60:5 (2022) 5-17.
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (18,1).

Cf. A010533, A020839, A040072, A041144.
Cf. similar sequences listed in A243399.
Row n=18 of A073133, A172236 and A352361 and column k=18 of A157103.

[n le 2 select (18)^(n-1) else 18*Self(n-1)+Self(n-2): n in [1..30]]; // G. C. Greubel, Sep 29 2024

Denominator[Convergents[Sqrt[82], 30]] (* Vincenzo Librandi, Dec 11 2013 *)
Fibonacci[Range[30], 18] (* G. C. Greubel, Sep 29 2024 *)

A041145=BinaryRecurrenceSequence(18,1,1,18)
[A041145(n) for n in range(31)] # G. C. Greubel, Sep 29 2024

a(n) = Fibonacci(n+1, 18), the n-th Fibonacci polynomial evaluated at x=18. - T. D. Noe, Jan 19 2006
From Philippe Deléham, Nov 21 2008: (Start)
a(n) = 18*a(n-1) + a(n-2) for n > 1; a(0)=1, a(1)=18.
G.f.: 1/(1 - 18*x - x^2). (End)
E.g.f.: exp(9*x)*(cosh(sqrt(82)*x) + 9*sinh(sqrt(82)*x)/sqrt(82)). - Stefano Spezia, Oct 02 2024

A040072 Continued fraction for sqrt(82).

Original entry on oeis.org

9, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18

Offset: 0

N. J. A. Sloane

			9.05538513813741662657380... = 9 + 1/(18 + 1/(18 + 1/(18 + 1/(18 + ...)))). - _Harry J. Smith_, Jun 10 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 (1).

Cf. A010533 (decimal expansion), A041144/A041145 (convergents), A248305 (Egyptian fraction).

Cf. A040000, A040006.

Digits := 100: convert(evalf(sqrt(N)),confrac,90,'cvgts'):

ContinuedFraction[Sqrt[82],300] (* Vladimir Joseph Stephan Orlovsky, Mar 09 2011 *)
PadRight[{9},120,{18}] (* Harvey P. Dale, Oct 09 2020 *)

{ allocatemem(932245000); default(realprecision, 51000); x=contfrac(sqrt(82)); for (n=0, 20000, write("b040072.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 10 2009

From Elmo R. Oliveira, Feb 10 2024: (Start)
a(n) = 18 = A010857(n) for n >= 1.
G.f.: 9*(1+x)/(1-x).
E.g.f.: 18*exp(x) - 9.
a(n) = 9*A040000(n) = 3*A040006(n). (End)

A041144 Numerators of continued fraction convergents to sqrt(82).

Original entry on oeis.org

9, 163, 2943, 53137, 959409, 17322499, 312764391, 5647081537, 101960232057, 1840931258563, 33238722886191, 600137943210001, 10835721700666209, 195643128555201763, 3532412035694297943, 63779059771052564737, 1151555487914640463209, 20791777842234580902499

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (18,1).

Cf. A010533, A041145.

CoefficientList[Series[(9 + x)/(1 - 18 x - x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 29 2013 *)

From Philippe Deléham, Nov 21 2008: (Start)
a(n) = 18*a(n-1) + a(n-2), n > 1; a(0)=9, a(1)=163.
G.f.: (9+x)/(1-18*x-x^2). (End)

More terms from Colin Barker, Nov 05 2013

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A248305 Egyptian fraction representation of sqrt(82) (A010533) using a greedy function.

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A041145 Denominators of continued fraction convergents to sqrt(82).

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A040072 Continued fraction for sqrt(82).

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A041144 Numerators of continued fraction convergents to sqrt(82).

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