A222471 -id:A222471 - cp's OEIS frontend

A222469 Denominator sequence of the n-th convergent of the continued fraction 1/(1 - 2/(2 - 2/(3 - 2/(4 - ...)))).

1, 1, 0, -2, -8, -36, -200, -1328, -10224, -89360, -873152, -9425952, -111365120, -1428894656, -19781794944, -293869134848, -4662342567680, -78672085380864, -1406772851720192, -26571340011921920, -528613254534998016

Offset: 0

Gary Detlefs and Wolfdieter Lang, Mar 23 2013

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.

			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.

G. C. Greubel, Table of n, a(n) for n = 0..445

Cf. A084950, A221913, A222470, A222471.

Cf. A001040(n+1) (x=1), A058797 (x=-1), A222467 (x=2).

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

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 *)

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

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...

A222470 Numerator sequence of the n-th convergent of the continued fraction 1/(1-2/(2-2/(3-2/(4-...

Original entry on oeis.org

1, 2, 4, 12, 52, 288, 1912, 14720, 128656, 1257120, 13571008, 160337856, 2057250112, 28480825856, 423097887616, 6712604550144, 113268081577216, 2025400259289600, 38256068763347968, 761070574748380160

Offset: 1

Gary Detlefs and Wolfdieter Lang, Mar 23 2013

The corresponding denominator sequence is A222469(n).
a(n) = Phat(n,-2) with the numerator polynomials Phat of A221913. All the given formulas follow from there and the comments given under A084950. The limit of the continued fraction (0 + K_{k=1..oo} (-2/k))/(-2) = 1/(1-2/(2-2/(3-2/(4-... is (1/2)*sqrt(2)*BesselJ(1,2*sqrt(2))/BesselJ(0,2*sqrt(2)) = -1.43974932187023280... (see A222471).
For a combinatorial interpretation in terms of labeled Morse codes see a comment on A221913. Here each dash has label x=-2, and the dots have label j if they are at position j. Labels are multiplied and all codes on [2,...,n+1] are summed.

			a(4) = 4*a(3) - 2*a(2) = 4*4 - 2*2 = 12.
Continued fraction convergent: 1/(1-2/(2-2/(3-2/4))) = -3/2 = 12/(-8) = a(4)/A222469(4).
Morse code a(5) = 52 from the sum of all 5 labeled codes on [2,3,4,5], one with no dash, three with one dash and one with two dashes: 5!/1 + (4*5 + 2*5 + 2*3)*(-2) +(-2)^2 = 52.

Cf. A084950, A221913, A222467, A001040(n+1) (x=1), A058798 (x=-1), A222468 (x=2).

Rest[RecurrenceTable[{a[-1]==-(1/2),a[0]==0,a[n]==n*a[n-1]-2a[n-2]},a,{n,20}]] (* Harvey P. Dale, Oct 24 2015 *)

Recurrence: a(n) = n*a(n-1) - 2*a(n-2), a(-1) = -1/2, a(0) = 0, n >= 1.
As a sum: a(n) = Sum_{m=0..floor(n/2)} a(n-m,m)*(-2)^m, n >= 1, with a(n,m) = binomial(n,m)*(n+1)!/(m+1)! = |A066667(n,m)| (Laguerre coefficients, parameter alpha = 1).
Explicit form: a(n) = Pi*(z/2)^n*(BesselY(1,z)*BesselJ(n+1,z) - BesselJ(1,z)*BesselY(n+1,z)) with z = 2*sqrt(2).
E.g.f.: Pi*(BesselJ(1, -x*sqrt(1-z))*BesselY(1, -x) - BesselY(1, -x*sqrt(1-z))*BesselJ(1, -x))/sqrt(1-z) with x = 2*sqrt(x). Here Phat(0,x) = 0.
Asymptotics: lim_{n->oo} a(n)/n! = BesselJ(1,2*sqrt(2))/(sqrt(2)) = 0.2829799868805...

cp's OEIS Frontend

A222469 Denominator sequence of the n-th convergent of the continued fraction 1/(1 - 2/(2 - 2/(3 - 2/(4 - ...)))).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula