cp's OEIS Frontend

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.

Showing 1-3 of 3 results.

A058798 a(n) = n*a(n-1) - a(n-2) with a(0) = 0, a(1) = 1.

Original entry on oeis.org

0, 1, 2, 5, 18, 85, 492, 3359, 26380, 234061, 2314230, 25222469, 300355398, 3879397705, 54011212472, 806288789375, 12846609417528, 217586071308601, 3903702674137290, 73952764737299909, 1475151592071860890
Offset: 0

Views

Author

Christian G. Bower, Dec 02 2000

Keywords

Comments

Note that a(n) = (a(n-1) + a(n+1))/(n+1). - T. D. Noe, Oct 12 2012; corrected by Gary Detlefs, Oct 26 2018
a(n) = log_2(A073888(n)) = log_3(A073889(n)).
a(n) equals minus the determinant of M(n+2) where M(n) is the n X n symmetric tridiagonal matrix with entries 1 just above and below its diagonal and diagonal entries 0, 1, 2, .., n-1. Example: M(4)=matrix([[0, 1, 0, 0], [1, 1, 1, 0], [0, 1, 2, 1], [0, 0, 1, 3]]). - Roland Bacher, Jun 19 2001
a(n) = A221913(n,-1), n>=1, is the numerator sequence of the n-th approximation of the continued fraction -(0 + K_{k>=1} (-1/k)) = 1/(1-1/(2-1/(3-1/(4-... The corresponding denominator sequence is A058797(n). - Wolfdieter Lang, Mar 08 2013
The recurrence equation a(n+1) = (A*n + B)*a(n) + C*a(n-1) with the initial conditions a(0) = 0, a(1) = 1 has the solution a(n) = Sum_{k = 0..floor((n-1)/2)} C^k*binomial(n-k-1,k)*( Product_{j = 1..n-2k-1} (k+j)*A + B ). This is the case A = 1, B = 1, C = -1. - Peter Bala, Aug 01 2013

Examples

			Continued fraction approximation 1/(1-1/(2-1/(3-1/4))) = 18/7 = a(4)/A058797(4). - _Wolfdieter Lang_, Mar 08 2013

Links

Crossrefs

Column 1 of A007754.
Cf. A073888, A073889, A221913 (alternating row sums).
Other recurrences of this type: A001040, A036242, A036244, A053983, A053984, A053987, A058307, A058308, A058309, A058797, A058799, A075374, A106174, A121323, A121351, A121353, A121354, A222468, A222470.
Similar sequences: A000806, A001053, A007754, A025164, A093986, A159927, A222467, A222469.

Programs

  • GAP
    a:=[1,2];; for n in [3..25] do a[n]:=n*a[n-1]-a[n-2]; od; Concatenation([0], a); # Muniru A Asiru, Oct 26 2018
  • Magma
    [0] cat [n le 2 select n else n*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Sep 22 2016
  • Mathematica
    t = {0, 1}; Do[AppendTo[t, n*t[[-1]] - t[[-2]]], {n, 2, 25}]; t (* T. D. Noe, Oct 12 2012 *)
nxt[{n_,a_,b_}]:={n+1,b,b*(n+1)-a}; Transpose[NestList[nxt,{1,0,1},20]] [[2]] (* Harvey P. Dale, Nov 30 2015 *)
  • PARI
    m=30; v=concat([1,2], vector(m-2)); for(n=3, m, v[n] = n*v[n-1]-v[n-2]); concat(0, v) \\ G. C. Greubel, Nov 24 2018
  • Sage
    def A058798(n):
    if n < 3: return n
    return hypergeometric([1/2-n/2, 1-n/2],[2, 1-n, -n], -4)*factorial(n)
[simplify(A058798(n)) for n in (0..20)] # Peter Luschny, Sep 10 2014

Formula

a(n) = Sum_{k = 0..floor((n-1)/2)} (-1)^k*binomial(n-k-1,k)*(n-k)!/(k+1)!. - Peter Bala, Aug 01 2013
a(n) = A058797(n+1) + A058799(n-1). - Henry Bottomley, Feb 28 2001
a(n) = Pi*(BesselY(1, 2)*BesselJ(n+1, 2) - BesselJ(1,2)* BesselY(n+1,2)). See the Abramowitz-Stegun reference given under A103921, p. 361 eq. 9.1.27 (first line with Y, J and z=2) and p. 360, eq. 9.1.16 (Wronskian). - Wolfdieter Lang, Mar 05 2013
Limit_{n->oo} a(n)/n! = BesselJ(1,2) = 0.576724807756873... See a comment on asymptotics under A084950.
a(n) = n!*hypergeometric([1/2-n/2, 1-n/2], [2, 1-n, -n], -4) for n >= 2. - Peter Luschny, Sep 10 2014

Extensions

New description from Amarnath Murthy, Aug 17 2002

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...

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

Views

Author

Gary Detlefs and Wolfdieter Lang, Mar 23 2013

Keywords

Comments

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.

Examples

			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.

Crossrefs

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

Programs

  • Mathematica
    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 *)

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,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...
Showing 1-3 of 3 results.

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