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A041143 Denominators of continued fraction convergents to sqrt(80). 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

LINKS

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).

FORMULA

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

MAPLE

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

MATHEMATICA

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 *)

PROG

(MAGMA) 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

(PARI) 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

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

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

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

(Python) 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

CROSSREFS

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

Sequence in context: A041606 A140142 A197351 * A041608 A041609 A041610

Adjacent sequences:  A041140 A041141 A041142 * A041144 A041145 A041146

KEYWORD

nonn,frac,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

STATUS

approved

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Last modified December 12 15:11 EST 2019. Contains 329960 sequences. (Running on oeis4.)