A040071 -id:A040071 - cp's OEIS frontend

A010532 Decimal expansion of square root of 80.

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

Offset: 1

N. J. A. Sloane

Continued fraction expansion is 8 followed by {1, 16} repeated. - Harry J. Smith, Jun 09 2009

			8.944271909999158785636694674925104941762473438446102897083588981642083....

Harry J. Smith, Table of n, a(n) for n = 1..20000
Index entries for algebraic numbers, degree 2

Cf. A040071 (continued fraction). - Harry J. Smith, Jun 09 2009

RealDigits[Sqrt[80],10,120][[1]] (* Harvey P. Dale, Oct 20 2011 *)

default(realprecision, 20080); x=sqrt(80); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010532.txt", n, " ", d));  \\ Harry J. Smith, Jun 09 2009

From Amiram Eldar, Aug 02 2020: (Start)
Equals 10 * 2/sqrt(5) = 10 * Sum_{k>=0} (-1)^k * binomial(2*k,k)/16^k.
Equals 8 + Sum_{k>=0} (-1)^k * binomial(2*k,k)/((k+1) * 16^k). (End)

Final digits of sequence corrected using the b-file. - N. J. A. Sloane, Aug 30 2009

A067280 Number of terms in continued fraction for sqrt(n), excl. 2nd and higher periods.

Original entry on oeis.org

1, 2, 3, 1, 2, 3, 5, 3, 1, 2, 3, 3, 6, 5, 3, 1, 2, 3, 7, 3, 7, 7, 5, 3, 1, 2, 3, 5, 6, 3, 9, 5, 5, 5, 3, 1, 2, 3, 3, 3, 4, 3, 11, 9, 7, 13, 5, 3, 1, 2, 3, 7, 6, 7, 5, 3, 7, 8, 7, 5, 12, 5, 3, 1, 2, 3, 11, 3, 9, 7, 9, 3, 8, 6, 5, 13, 7, 5, 5, 3, 1, 2, 3, 3, 6, 11, 3, 7, 6, 3, 9, 9, 11, 17, 5, 5, 12, 5

Offset: 1

Frank Ellermann, Feb 23 2002

			a(2)=2: [1,(2)+ ]; a(3)=3: [1,(1,2)+ ]; a(4)=1: [2]; a(5)=2: [2,(4)+ ].

H. Davenport, The Higher Arithmetic. Cambridge Univ. Press, 7th edition, 1999, table 1.

Related sequences: 2 : A040000, ..., 44: A040037, 48: A040041, ..., 51: A040043, 56: A040048, 60: A040052, 63: A040055, ..., 66: A040057. 68: A040059, 72: A040063, 80: A040071.
Related sequences: 45: A010135, ..., 47: A010137, 52: A010138, ..., 55: A010141, 57: A010142, ..., 59: A010144. 61: A010145, 62: A010146. 67: A010147, 69: A010148, ..., 71: A010150.
Cf. A003285.

from sympy import continued_fraction_periodic
def A067280(n): return len((a := continued_fraction_periodic(0,1,n))[:1]+(a[1] if a[1:] else [])) # Chai Wah Wu, Jun 14 2022

a(n) = A003285(n) + 1. - Andrey Zabolotskiy, Jun 23 2020

Name clarified by Michel Marcus, Jun 22 2020

A041143 Denominators of continued fraction convergents to sqrt(80).

Original entry on oeis.org

1, 1, 17, 18, 305, 323, 5473, 5796, 98209, 104005, 1762289, 1866294, 31622993, 33489287, 567451585, 600940872, 10182505537, 10783446409, 182717648081, 193501094490, 3278735159921, 3472236254411, 58834515230497, 62306751484908, 1055742538989025

Offset: 0

N. J. A. Sloane

This is the sequence of Lehmer numbers U_n(sqrt(R),Q) for the parameters R = 16 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..199 [First term 0 removed by _Georg Fischer_, Jul 01 2019]
Eric W. Weisstein, MathWorld: Lehmer Number
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (0,18,0,-1).

Cf. A041142, A020837, A040071, A010532, A001076, A002530.

List([0..30], n-> (5 +3*(-1)^n)*Fibonacci(3*(n+1))/16 ); # G. C. Greubel, Jul 02 2019

I:=[1,1,17,18]; [n le 4 select I[n] else 18*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 11 2013

with(numtheory): cf := cfrac(sqrt(80),25): seq(nthdenom(cf,n), n=0..24); # Peter Luschny, Jul 06 2019

Denominator/@Convergents[Sqrt[80], 30] (* Vladimir Joseph Stephan Orlovsky, Jul 05 2011 *)
CoefficientList[Series[(1 + x - x^2)/(1 - 18 x^2 + x^4), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 11 2013 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,0,18,0]^n*[1;1;17;18])[1,1] \\ Charles R Greathouse IV, Nov 13 2015

a(n) = (5 + 3*(-1)^n)*fibonacci(3*(n+1))/16 \\ Georg Fischer, Jul 01 2019

from sympy import continued_fraction_convergents as convergents, continued_fraction_iterator as cf, sqrt, denom
denominators = (denom(c) for c in convergents(cf(sqrt(80))))
print([next(denominators) for  in range(30)]) # _Ehren Metcalfe, Jul 03 2019

[(5 +3*(-1)^n)*fibonacci(3*(n+1))/16 for n in (0..30)] # G. C. Greubel, Jul 02 2019

G.f.: (1 + x - x^2) / (1 - 18*x^2 + x^4).
a(n) = 18*a(n-2) - a(n-4).
From Peter Bala, May 28 2014: (Start)
Let alpha = 2 + sqrt(5) and beta = 2 - sqrt(5) be the roots of the equation x^2 - 4*x - 1 = 0. Then a(n) = (alpha^n - beta^n)/(alpha - beta) for n even, while a(n) = (alpha^n - beta^n)/(alpha^2 - beta^2) for n odd.
a(n) = A001076(n+1) for n even; a(n) = 1/4*A001076(n+1) for n odd.
a(n) = Product_{k = 1..floor(n/2)} ( 16 + 4*cos^2(k*Pi/(n+1)) ).
Recurrence equations: a(0) = 1, a(1) = 1 and for n >= 1, a(2*n) = 16*a(2*n - 1) + a(2*n - 2) and a(2*n + 1) = a(2*n) + a(2*n - 1). (End)
a(n) = (5 + 3*(-1)^n)*Fibonacci(3*(n+1))/16. - Ehren Metcalfe, Apr 15 2019

First term 0 removed from b-file, formulas and programs by Georg Fischer, Jul 01 2019

A267319 Continued fraction expansion of phi^8, where phi = (1 + sqrt(5))/2.

Original entry on oeis.org

46, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1

Offset: 0

Ilya Gutkovskiy, Jan 13 2016

More generally, the ordinary generating function for the continued fraction expansion of phi^(2*k + 1), where phi = (1 + sqrt(5))/2, k = 1, 2, 3,... is floor(phi^(2*k + 1))/(1 - x), and for the continued fraction expansion of phi^(2*k) is (floor(phi^(2*k)) + x - x^2)/(1 - x^2).

			phi^8 = (47 + 21*sqrt(5))/2 = 46 + 1/(1 + 1/(45 + 1/(1 + 1/(45 + 1/(1 + 1/(45 + 1/...)))))).

Eric Weisstein's World of Mathematics, Golden Ratio
Wikipedia, Golden ratio
Index entries for continued fractions for constants
Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A001622.

Cf. continued fraction expansion of phi^k: A000012 (k = 1), A054977 (k = 2), A010709 (k = 3), A176260 (k = 4, for n>0), A010850 (k = 5), A040071 (k = 6, for n>0), A010868 (k = 7), this sequence (k = 8).

[46] cat &cat [[1, 45]^^50]; // Vincenzo Librandi, Jan 13 2016

ContinuedFraction[(47 + 21 Sqrt[5])/2, 82]

G.f.: (46 + x - x^2)/(1 - x^2).

a(n) = 23 + 22*(-1)^n for n>0. - Bruno Berselli, Jan 18 2016

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A010532 Decimal expansion of square root of 80.

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A067280 Number of terms in continued fraction for sqrt(n), excl. 2nd and higher periods.

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A041143 Denominators of continued fraction convergents to sqrt(80).

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A267319 Continued fraction expansion of phi^8, where phi = (1 + sqrt(5))/2.

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