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A136211
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Denominators in continued fraction [0; 1, 3, 1, 3, 1, 3, ...].
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6
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1, 4, 5, 19, 24, 91, 115, 436, 551, 2089, 2640, 10009, 12649, 47956, 60605, 229771, 290376, 1100899, 1391275, 5274724, 6665999, 25272721, 31938720, 121088881, 153027601, 580171684, 733199285, 2779769539, 3512968824
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OFFSET
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1,2
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COMMENTS
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A136210(n)/A136211(n) tends to 0.791287847... = [0; 1, 3, 1, 3, 1, 3, ...] = (sqrt(21) - 3)/2 = the inradius of a right triangle with hypotenuse 3, legs 2 and sqrt(21).
The number 0.791287847... = (sqrt(21) - 3)/2 arises in finding a number which is 5 less than its square; the result is: 2.791287847... because (2.791287847...)^2 = 7.791287847... In general the quadratic equation for finding such numbers is x^2 - x = N, so x = (1 + sqrt(1 + 4N))/2. - Alexander R. Povolotsky, Dec 23 2007
Prepending a 1 to the sequence gives [1, 1, 4, 5, 19, 24, ...]. This is the sequence of Lehmer numbers U_n(sqrt(R),Q) with the parameters R = 3 and Q = -1. It is a strong divisibility sequence, that is, GCD(a(n),a(m)) = a(GCD(n,m)) for all natural numbers n and m. - Peter Bala, May 14 2014
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LINKS
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FORMULA
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a(1) = 1, a(2) = 4, then for n>2, a(2n) = 3*a(2n-1) + a(2n-2); a(2n-1) = a(2n-2) + a(2n-3). Let T = the 2 X 2 matrix [1, 3; 1, 4]. Then T^n = [A136210(2n-1), A136210(2n); a(2n-1), a(2n)].
O.g.f.: x*(1+4*x-x^3)/(1-5*x^2+x^4).
{-a(n) + 5 a(n + 2) - a(n + 4), a(0) = 1, a(1) = 4, a(2) = 5, a(3) = 19}. - Robert Israel, May 14 2008
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EXAMPLE
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a(4) = 19 = 3*a(3) + a(2) = 3*5 + 4.
a(5) = 24 = a(4) + a(3) = 19 + 5.
T^3 = [19, 72; 24, 91], where the bottom row [24, 91] = [a(5), a(6)].
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MATHEMATICA
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a[n_] := FromContinuedFraction[ Join[{0}, 3 - 2*Array[Mod[#, 2]&, n]]] // Denominator; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, May 15 2014 *)
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PROG
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(PARI) x='x + O('x^25); Vec(x*(1+4*x-x^3)/(1-5*x^2+x^4)) \\ G. C. Greubel, Feb 18 2017
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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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STATUS
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approved
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