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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A222467 Denominator sequence of the n-th convergent of the continued fraction 1/(1 + 2/(2 + 2/(3 + 2/(4 + ...

Original entry on oeis.org

1, 1, 4, 14, 64, 348, 2216, 16208, 134096, 1239280, 12660992, 141749472, 1726315648, 22725602368, 321611064448, 4869617171456, 78557096872192, 1345209881170176, 24370892054807552, 465737368803683840, 9363489160183291904
Offset: 0

Views

Author

Gary Detlefs and Wolfdieter Lang, Mar 21 2013

Keywords

Comments

The corresponding numerator sequence is A222468.
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 (0 + K_{k>=1} (2/k))/2 = 1/(1+2/(2+2/(3+2/(4+... is (1/2)*sqrt(2)*BesselI(1,2*sqrt(2))/BesselI(0,2*sqrt(2)) = 0.5631786198117... See A222466 for more digits.
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 all codes on [1,2,...,n] are summed.

Examples

			a(4) = 4*a(3) + 2*a(2) = 4*14 + 2*4 = 64.
Continued fraction convergent: 1/(1+2/(2+2/(3+2/4))) = 9/16 =  36/64 = A222468(4)/a(4).
Morse code a(4) = 64 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 = 64.

Links

Crossrefs

Cf. A084950, A221913, A222468, A001040(n+1) (x=1), A058797 (x=-1).

Programs

  • Magma
    I:=[1,4]; [1] cat [n le 2 select I[n] else n*Self(n -1) + 2*Self(n-2): n in [1..30]]; // G. C. Greubel, May 16 2018
  • Mathematica
    RecurrenceTable[{a[0]==a[1]==1,a[n]==n*a[n-1]+2*a[n-2]},a,{n,20}] (* Harvey P. Dale, Jul 06 2017 *)
  • PARI
    m=30; v=concat([1,4], vector(m-2)); for(n=3, m, v[n]=n*v[n-1] + 2*v[n-2]); concat([1], v) \\ G. C. Greubel, May 16 2018

Formula

Recurrence: a(n) = n*a(n-1) + 2*a(n-2), a(-1) = 0, a(0 ) =1, n >= 1.
As a sum: 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).
Explicit form: a(n) = 2*(w/2)^(n+1)*(BesselI(0,w)*BesselK(n+1,w) - BesselK(0,w)*BesselI(n+1,w)*(-1)^(n+1)) with w := -2*sqrt(2).
E.g.f.: (i*Pi*sqrt(2)/sqrt(1-z))*(BesselJ(1, 2*i*sqrt(2)*sqrt(1-z))*BesselY(0, 2*i*sqrt(2)) - BesselY(1, 2*i*sqrt(2)*sqrt(1-z))*BesselJ(0,2*i*sqrt(2))) with the imaginary unit i = sqrt(-1).
Asymptotics: lim_{n->infinity} a(n)/n! = BesselI(0,2*sqrt(2)) = 4.2523508795026...

A222468 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, 8, 36, 196, 1248, 9128, 75520, 697936, 7130400, 79830272, 972224064, 12798573376, 181124475392, 2742464277632, 44241677392896, 757593444234496, 13725165351006720, 262293328557596672, 5273316901853946880
Offset: 1

Views

Author

Gary Detlefs and Wolfdieter Lang, Mar 21 2013

Keywords

Comments

The corresponding denominator sequence is A222467(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} (2/k))/2 = 1/(1+2/(2+2/(3+2/(4+... is (1/2)*sqrt(2)*BesselI(1,2*sqrt(2))/BesselI(0,2*sqrt(2)) = 0.5631786198117... See A222466 for more decimals.
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.

Examples

			a(4) = 4*a(3) + 2*a(2) = 4*8 + 2*2 = 36.
Continued fraction convergent: 1/(1+2/(2+2/(3+2/4))) = 9/16 = 36/64 = a(4)/A222467(4).
Morse code a(5) = 196 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 = 196.

Links

Crossrefs

Cf. A084950, A221913, A222467, A001040(n+1) (x=1), A058797 (x=-1).

Programs

  • Magma
    I:=[1, 2]; [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
  • Mathematica
    RecurrenceTable[{a[1] == 1, a[2] == 2, a[n] == n*a[n - 1] + 2 a[n - 2]}, a[n], {n, 20}] (* Ray Chandler, Jul 30 2015 *)
  • PARI
    m=30; v=concat([1,2], vector(m-2)); for(n=3, m, v[n]=n*v[n-1] +2*v[n-2]); v \\ G. C. Greubel, May 17 2018

Formula

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-1-m,m)*(n-m)!/(m+1)! = |A066667(n,m)| (Laguerre coefficients, parameter alpha = 1).
Explicit form: a(n) = 2*(w/2)^n*(BesselI(1,w)*BesselK(n+1,w) + BesselK(1,w)*BesselI(n+1,w)*(-1)^(n+1)), n >= 1, where w := -2*sqrt(2).
E.g.f.: Pi*(BesselJ(1, 2*i*sqrt(2)*sqrt(1-z))* BesselY(1, 2*i*sqrt(2)) - BesselY(1, (2*i)*sqrt(2)*sqrt(1-z))*BesselJ(1, 2*i*sqrt(2)))/sqrt(1-z) with Bessel functions and the imaginary unit i = sqrt(-1). Phat(0,x) = 0.
Asymptotics: lim_{n -> infinity} a(n)/n! = BesselI(1,2*sqrt(2)) /sqrt(2) = 2.3948330992734...

A222471 Decimal expansion of the negative of the limit of the continued fraction 1/(1-2/(2-2/(3-2/(4-... in terms of Bessel functions.

Original entry on oeis.org

1, 4, 3, 9, 7, 4, 9, 3, 2, 1, 8, 7, 0, 2, 3, 2, 8, 0, 5, 8, 9, 5, 7, 0, 6, 9, 5, 7, 4, 1, 1, 2, 2, 7, 4, 2, 5, 1, 5, 2, 7, 3, 3, 7, 6, 2, 2, 3, 8, 1, 1, 6, 1, 7, 5, 2, 8, 1, 4, 5, 3, 0, 7, 8, 8, 7, 7, 2, 3, 6, 1, 6, 8, 1, 6, 4, 3, 4, 5, 9, 6, 3, 8, 5, 0, 1, 9, 5, 1, 3, 1, 8, 5, 9, 7, 7, 0, 4, 8, 7, 6, 3, 4, 1, 7, 8, 7, 4, 0, 2
Offset: 1

Views

Author

Wolfdieter Lang, Mar 23 2013

Keywords

Comments

The continued fraction (0 + K_{k=1..oo} (-2/k))/(-2) = 1/(1-2/(2-2/(3-2/(4- ... converges, and its negative limit is given in the formula section in terms of Bessel functions.
In general, the continued fraction 0 + K_{k=1..oo} (x/k) = x/(1+x/(2+x/(3+... has n-th approximation x*Phat(n,x)/Q(n,x), with the row polynomials Phat of A221913 and Q of A084950. These polynomials are written in terms of Bessel function. Divided by n! = Gamma(n+1) one knows the limit for n -> infinity for these two polynomial systems for given x. This results in the formula 0 + K_{k=1..oo} (x/k) = sqrt(x)*BesselI(1,2*sqrt(x))/BesselI(0,2*sqrt(x)).
For x=1 see for the limit of the continued fraction A052119 and for the n-th approximation A001053(n+1)/A001040(n+1).

Examples

			-1.4397493218702328058...

Links

Crossrefs

Cf. A052119 (x=1), A222466 (x=2), A222469/A222470.

Programs

  • Mathematica
    RealDigits[BesselJ[1, 2*Sqrt[2]]/(Sqrt[2]*BesselJ[0, 2*Sqrt[2]]), 10, 50][[1]] (* G. C. Greubel, Aug 16 2017 *)
  • PARI
    besselj(1,sqrt(8))/besselj(0,sqrt(8))/sqrt(2) \\ Charles R Greathouse IV, Feb 19 2014

Formula

Equals (1/2)*sqrt(2)*BesselJ(1,2*sqrt(2))/BesselJ(0,2*sqrt(2)).
Showing 1-3 of 3 results.

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