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A041143
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Denominators of continued fraction convergents to sqrt(80).
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2
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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
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OFFSET
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0,3
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COMMENTS
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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
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LINKS
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FORMULA
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G.f.: (1 + x - x^2) / (1 - 18*x^2 + x^4).
a(n) = 18*a(n-2) - a(n-4).
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) = 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
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MAPLE
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with(numtheory): cf := cfrac(sqrt(80), 25): seq(nthdenom(cf, n), n=0..24); # Peter Luschny, Jul 06 2019
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MATHEMATICA
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CoefficientList[Series[(1 + x - x^2)/(1 - 18 x^2 + x^4), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 11 2013 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,frac,easy
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AUTHOR
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EXTENSIONS
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First term 0 removed from b-file, formulas and programs by Georg Fischer, Jul 01 2019
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STATUS
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approved
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