A010516 -id:A010516 - cp's OEIS frontend

A248288 Egyptian fraction representation of sqrt(63) (A010516) using a greedy function.

7, 2, 3, 10, 256, 69688, 5330178475, 685643579227613855733, 19857919470304339362673575257858955364290957, 4322562711957148145852339662715119494243446939653452977452988955819120724647597129517346

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[ 63]]

A041111 Denominators of continued fraction convergents to sqrt(63).

Original entry on oeis.org

1, 1, 15, 16, 239, 255, 3809, 4064, 60705, 64769, 967471, 1032240, 15418831, 16451071, 245733825, 262184896, 3916322369, 4178507265, 62415424079, 66593931344, 994730462895, 1061324394239, 15853271982241

Offset: 0

N. J. A. Sloane

The following remarks assume an offset of 1. This is the sequence of Lehmer numbers U_n(sqrt(R),Q) for the parameters R = 14 and Q = -1; it is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for all positive integers n and m. Consequently, this is a divisibility sequence: if n divides m then a(n) divides a(m). - Peter Bala, May 28 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Eric W. Weisstein, MathWorld: Lehmer Number
Index entries for linear recurrences with constant coefficients, signature (0,16,0,-1).

Cf. A010516, A041110, A002530.

Denominator/@Convergents[Sqrt[63],30] (* Harvey P. Dale, May 18 2011 *)
CoefficientList[Series[(1 + x - x^2)/(1 - 16 x^2 + x^4), {x, 0, 30}], x]  (* Vincenzo Librandi, Oct 24 2013 *)

From Colin Barker, Jul 15 2012: (Start)
a(n) = 16*a(n-2) - a(n-4).
G.f.: (1+x-x^2)/(1-16*x^2+x^4). (End)
From Peter Bala, May 28 2014: (Start)
The following remarks assume an offset of 1.
Let alpha = ( sqrt(14) + sqrt(18) )/2 and beta = ( sqrt(14) - sqrt(18) )/2 be the roots of the equation x^2 - sqrt(14)*x - 1 = 0. Then a(n) = (alpha^n - beta^n)/(alpha - beta) for n odd, while a(n) = (alpha^n - beta^n)/(alpha^2 - beta^2) for n even. a(n) = Product_{k = 1..floor((n-1)/2)} ( 14 + 4*cos^2(k*Pi/n) ). Recurrence equations: a(0) = 0, a(1) = 1 and for n >= 1, a(2*n) = a(2*n - 1) + a(2*n - 2) and a(2*n + 1) = 14*a(2*n) + a(2*n - 1). (End)

A040055 Continued fraction for sqrt(63).

Original entry on oeis.org

7, 1, 14, 1, 14, 1, 14, 1, 14, 1, 14, 1, 14, 1, 14, 1, 14, 1, 14, 1, 14, 1, 14, 1, 14, 1, 14, 1, 14, 1, 14, 1, 14, 1, 14, 1, 14, 1, 14, 1, 14, 1, 14, 1, 14, 1, 14, 1, 14, 1, 14, 1, 14, 1, 14, 1, 14, 1, 14, 1, 14, 1, 14, 1, 14, 1

Offset: 0

N. J. A. Sloane

			7.9372539331937717715048472... = 7 + 1/(1 + 1/(14 + 1/(1 + 1/(14 + ...)))). - _Harry J. Smith_, Jun 07 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 (0,1).

Cf. A010516 (decimal expansion), A020820 (decimal expansion of 1/sqrt(63)).

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

ContinuedFraction[Sqrt[63], 300] (* Vladimir Joseph Stephan Orlovsky, Mar 08 2011 *)

{ allocatemem(932245000); default(realprecision, 25000); x=contfrac(sqrt(63)); for (n=0, 20000, write("b040055.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 07 2009

From Amiram Eldar, Nov 13 2023: (Start)
Multiplicative with a(2^e) = 14, and a(p^e) = 1 for p >= 5.
Dirichlet g.f.: zeta(s) * (1 + 13/2^s). (End)

A377291 For each row n in array A374602(n, k), the asymptotic geometric growth factor of every A377290(n) terms, represented by its nearest integer.

Original entry on oeis.org

6, 14, 7, 98, 16, 34, 1442, 398, 194, 119, 30, 62, 4354, 1154, 115598, 322, 23, 155234, 48, 98, 10402, 2702, 64514, 727, 482, 3040, 1154, 2114, 70, 142, 21314, 5474, 2498, 1442, 16793602, 674, 48497294, 158402, 47, 48670, 96, 194, 39202, 9998, 1684802, 2599

Offset: 1

Charles L. Hohn, Oct 23 2024

nonn

(a(n)^2-4)/A000037(n) is a square, and as such, a(n) is a member of row x of A298675(x, k), where x is the smallest value >= 3 such that (x^2-4)/A000037(n) is a square. E.g. for n=38: A000037(38)=44, x=20 ((20^2-4)/44 = 3^2), and a(38) = 158402 = A298675(20, 4).

The same row x of A298675(x, k) also results as integer solutions to g+(1/g) where g=(w*sqrt(d) + ceiling(w*sqrt(d)))/2 and d=A000037(n) for integers w >= 1. As such, it follows that g(n) can be expressed as a simple integer arithmetic transformation of sqrt(A000037(n)), e.g. g(1) = 2*sqrt(2)+3 (A156035), g(2) = 4*sqrt(3)+7 (A354129), g(3) = (3*sqrt(5)+7)/2 (A374883), g(4) = 20*sqrt(6)+49, and g(5) = 3*sqrt(7)+8 (A010516+8).

			For n = 5, the first few terms of A374602(5, k) are {4, 5, 11, 28, 62, 79, 175, 446, 988} and the period size is A377290(5) = 4, giving A374602(5, 1+4)/A374602(5, 1) = 62/4 = 15.5, 79/5 = 15.8, 175/11 = 15.909..., 446/28 = 15.928..., 988/62 = 15.935..., ..., to limit 15.937... -> g(5), from which g(5)+(1/g(5)) = 16 -> a(5).

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

Cf. A374602, A377290, A000037, A298675.

Cf. A156035, A354129, A374883, A010516.

Growth factor g(n) = Lim_{k->oo}(A374602(n, k+A377290(n))/A374602(n, k)).
a(n) = round(g(n)) = ceiling(g(n)) = g(n)+(1/g(n)).
Inverse: g(n) = (sqrt(a(n)^2-4)+a(n))/2.
For d = A000037(n) and x in {1, 2, 4}, when d+x is a square (unless x==4 and d+x is even): a(n) = 4/x*d+2.
For d = A000037(n) and x in {-4, 1, 2, 4}, when n > 3 and d-x is a square (unless x==-4 and d-x is odd): a(n) = (4/abs(x))^2*d^2-16/x*d+2.

A020820 Decimal expansion of 1/sqrt(63).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

			0.1259881576697424090715055120780600202719171039563071...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A010516, A040055.

Digits:=100: evalf(1/sqrt(63)); # Wesley Ivan Hurt, Sep 01 2016

RealDigits[Sqrt[1/63],10,120][[1]] (* Harvey P. Dale, Jun 14 2011 *)

1/sqrt(63) = 1/(3*sqrt(7)) = sqrt(7)/21.

A041110 Numerators of continued fraction convergents to sqrt(63).

Original entry on oeis.org

7, 8, 119, 127, 1897, 2024, 30233, 32257, 481831, 514088, 7679063, 8193151, 122383177, 130576328, 1950451769, 2081028097, 31084845127, 33165873224, 495407070263, 528572943487, 7895428279081, 8424001222568

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0,16,0,-1).

Cf. A010516, A041111.

Numerator[Convergents[Sqrt[63], 30]] (* Vincenzo Librandi, Oct 26 2013 *)
LinearRecurrence[{0,16,0,-1},{7,8,119,127},30] (* Harvey P. Dale, Dec 17 2019 *)

a(n) = 16*a(n-2)-a(n-4). G.f.: (7+8*x+7*x^2-x^3)/(1-16*x^2+x^4). [Colin Barker, Jul 15 2012]

cp's OEIS Frontend

A248288 Egyptian fraction representation of sqrt(63) (A010516) using a greedy function.

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A041111 Denominators of continued fraction convergents to sqrt(63).

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A040055 Continued fraction for sqrt(63).

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A377291 For each row n in array A374602(n, k), the asymptotic geometric growth factor of every A377290(n) terms, represented by its nearest integer.

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A020820 Decimal expansion of 1/sqrt(63).

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A041110 Numerators of continued fraction convergents to sqrt(63).

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