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A222469
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Denominator sequence of the n-th convergent of the continued fraction 1/(1 - 2/(2 - 2/(3 - 2/(4 - ...)))).
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5
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1, 1, 0, -2, -8, -36, -200, -1328, -10224, -89360, -873152, -9425952, -111365120, -1428894656, -19781794944, -293869134848, -4662342567680, -78672085380864, -1406772851720192, -26571340011921920, -528613254534998016
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OFFSET
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0,4
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COMMENTS
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The corresponding numerator sequence is A222470(n).
a(n) = Q(n,-2) with the denominator polynomials Q of A084950. All the given formulas follow from there. The limit of the continued fraction (-1/2)*(0 + K_{k=1..oo} (-2/k) = 1/(1 - 2/(2 - 2/(3 - 2/(4 - ...)))) is (+1/2)*sqrt(2)*BesselJ(1,2*sqrt(2))/BesselJ(0,2*sqrt(2)) = -1.43974932187... For more decimals see A222471.
For a combinatorial interpretation in terms of labeled Morse codes see a comment on A084950. Here each dash has label x = -2, and the dots have label j if they are at position j. Labels are multiplied and for a(n) all labeled codes on [1,2,...,n] have to be summed.
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LINKS
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FORMULA
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a(n) = n*a(n-1) - 2*a(n-2), a(-1) = 0, a(0) = 1, n >= 1.
a(n) = Sum_{m=0..floor(n/2)} a(n-m, m)*(-2)^m, n >= 0, with a(n,m) = (n!/m!)*binomial(n,m) = |A021009(n,m)| (Laguerre).
a(n) = Pi*(z/2)^(n+1)*(BesselY(0,z)*BesselJ(n+1,z) - BesselJ(0,z)*BesselY(n+1,z)) with z := 2*sqrt(2).
E.g.f.: Pi*c/(2*sqrt(1-z))*(BesselJ(1, c*sqrt(1-z))*BesselY(0, c) - BesselY(1, c*sqrt(1-z))*BesselJ(0, c)), with c = 2*sqrt(2).
Asymptotics: lim_{n->oo} a(n)/n! = BesselJ(0, 2*sqrt(2)) = -0.1965480950...
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EXAMPLE
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a(4) = 4*a(3) - 2*a(2) = 4*(-2) + 2*0 = -8.
Continued fraction convergent: 1/(1 - 2/(2 - 2/(3 - 2/4))) = -3/2 = -12/8 = A222470(4)/a(4).
Morse code: a(4) = -8 from the sum of all 5 labeled codes on [1,2,3,4], one with no dash, three with one dash and one with two dashes: 4! + (3*4 + 1*4 + 1*2)*(-2) + (-2)^2 = -8.
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MATHEMATICA
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RecurrenceTable[{a[0] == 1, a[1] == 1, a[n] == n*a[n - 1] - 2 a[n - 2]}, a[n], {n, 50}] (* G. C. Greubel, Aug 16 2017 *)
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PROG
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(PARI) m=30; v=concat([1, 1], vector(m-2)); for(n=3, m, v[n]=n*v[n-1] -2*v[n-2]); v \\ G. C. Greubel, May 17 2018
(Magma) I:=[1, 1]; [n le 2 select I[n] else n*Self(n-1) - 2*Self(n-2): n in [1..30]]; // G. C. Greubel, May 17 2018
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CROSSREFS
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KEYWORD
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sign,easy,frac
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AUTHOR
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STATUS
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approved
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