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 1-4 of 4 results.

A157005 A Somos-4 variant.

Original entry on oeis.org

1, 2, 8, 24, 112, 736, 3776, 40192, 391424, 4203008, 85312512, 1270368256, 32235102208, 1038278549504, 27640704385024, 1549962593927168, 73624753456480256, 4273828146025070592, 435765959975516766208
Offset: 0

Views

Author

Paul Barry, Feb 20 2009

Keywords

Comments

Hankel transform of A157004.

Links

Crossrefs

Cf. A157101, A162546, A162547.

Programs

  • GAP
    a:=[1,2,8,24];; for n in [5..20] do a[n]:=(a[n-1]*a[n-3] + a[n-2]^2)/a[n-4]; od; a; # G. C. Greubel, Feb 23 2019
  • Magma
    I:=[1,2,8,24]; [n le 4 select I[n] else (Self(n-1)*Self(n-3) + Self(n-2)^2)/Self(n-4): n in [1..20]]; // G. C. Greubel, Feb 23 2019
  • Mathematica
    RecurrenceTable[{a[0]==1,a[1]==2,a[2]==8,a[3]==24,a[n]==(a[n-1] a[n-3]+a[n-2]^2)/a[n-4]},a,{n,20}]  (* Harvey P. Dale, Apr 30 2011 *)
  • PARI
    m=20; v=concat([1,2,8,24], vector(m-4)); for(n=5, m, v[n] = (v[n-1]*v[n-3] +v[n-2]^2)/v[n-4]); v \\ G. C. Greubel, Feb 23 2019
  • Sage
    def a(n):
    if (n==0): return 1
    elif (n==1): return 2
    elif (n==2): return 8
    elif (n==3): return 24
    else: return (a(n-1)*a(n-3) + a(n-2)^2)/a(n-4)
[a(n) for n in (0..20)] # G. C. Greubel, Feb 23 2019

Formula

a(n) = (a(n-1)*a(n-3) + a(n-2)^2)/a(n-4), with a(0)=1, a(1)=2, a(2)=8, a(3)=24.
a(n) = 2^n*A006720(n+2).

A157101 A Somos-4 variant.

Original entry on oeis.org

1, -1, -5, -4, 29, 129, -65, -3689, -16264, 113689, 2382785, 7001471, -398035821, -7911171596, 43244638645, 6480598259201, 124106986093951, -5987117709349201, -541051130050800400, -4830209396684261199
Offset: 0

Views

Author

Paul Barry, Feb 22 2009

Keywords

Comments

Hankel transform of A157100.

Links

Crossrefs

Cf. A051138.
Cf. A157005, A162546, A162547.

Programs

  • GAP
    a:=[1,-1,-5,-4];; for n in [5..20] do a[n]:=(a[n-1]*a[n-3] + a[n-2]^2)/a[n-4]; od; a; # G. C. Greubel, Feb 23 2019
  • Magma
    I:=[1,-1,-5,-4]; [n le 4 select I[n] else (Self(n-1)*Self(n-3) + Self(n-2)^2)/Self(n-4): n in [1..20]]; // G. C. Greubel, Feb 23 2019
  • Mathematica
    RecurrenceTable[{a[n]==(a[n-1]*a[n-3]+a[n-2]^2)/a[n-4], a[0]==1, a[1]==-1, a[2]==-5, a[3]==-4}, a, {n,20}] (* G. C. Greubel, Feb 23 2019 *)
  • PARI
    m=20; v=concat([1,-1,-5,-4], vector(m-4)); for(n=5, m, v[n] = (v[n-1]*v[n-3] +v[n-2]^2)/v[n-4]); v \\ G. C. Greubel, Feb 23 2019
  • Sage
    def a(n):
    if (n==0): return 1
    elif (n==1): return -1
    elif (n==2): return -5
    elif (n==3): return -4
    else: return (a(n-1)*a(n-3) + a(n-2)^2)/a(n-4)
[a(n) for n in (0..20)] # G. C. Greubel, Feb 23 2019

Formula

a(n) = (a(n-1)*a(n-3) + a(n-2)^2)/a(n-4), with a(0)=1, a(1)=-1, a(2)=-5, a(3)=-4.
a(n) = A051138(n+1) for all n in Z. - Michael Somos, Jul 17 2016

A162543 A Chebyshev transform of the large Schroeder numbers A006318.

Original entry on oeis.org

1, 2, 5, 18, 73, 312, 1391, 6406, 30235, 145478, 710951, 3519248, 17608681, 88914250, 452512229, 2318774506, 11953427329, 61948592936, 322570037543, 1686777086942, 8854240330363, 46638995523598, 246443050810895
Offset: 0

Views

Author

Paul Barry, Jul 05 2009

Keywords

Comments

Hankel transform is the Somos-4 variant A162546.

Links

Crossrefs

Cf. A162548.

Programs

  • GAP
    a:=[2,5,18,73,312,1391];; for n in [7..30] do a[n]:=(3*(2*n-1)*a[n-1] - (4*n-5)*a[n-2] +12*(n-2)*a[n-3] -(4*n-11)*a[n-4] +3*(2*n-7)*a[n-5] -(n-5)*a[n-6])/(n+1); od; Concatenation([1], a); # G. C. Greubel, Feb 23 2019
  • Magma
    m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1-x+x^2 - Sqrt(1-6*x+3*x^2-6*x^3+x^4))/( 2*x*(1+x^2)) )); // G. C. Greubel, Feb 23 2019
  • Mathematica
    CoefficientList[Series[(1-x+x^2 - Sqrt[1-6*x+3*x^2-6*x^3+x^4])/(2*x*(1+x^2)), {n,0,30}], x] (* G. C. Greubel, Feb 23 2019 *)
  • PARI
    my(x='x+O('x^30)); Vec((1-x+x^2 - sqrt(1-6*x+3*x^2-6*x^3+x^4))/( 2*x*(1+x^2))) \\ G. C. Greubel, Feb 23 2019
  • Sage
    ((1-x+x^2 -sqrt(1-6*x+3*x^2-6*x^3+x^4))/( 2*x*(1+x^2))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 23 2019

Formula

G.f.: (1/(1+x^2))*S(x/(1+x^2)), S(x) the g.f. of A006318;
G.f.: (1-x+x^2 - sqrt(1-6*x+3*x^2-6*x^3+x^4))/(2*x*(1+x^2)).
G.f.: 1/(1+x^2-2*x/(1-x/(1+x^2-2*x/(1-x/(1+x^2-2*x/(1-x/(1+x+2*x^2/(1-... (continued fraction);
a(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(n-k,k)*A006318(n-2*k).
Recurrence: (n+1)*a(n) = (5-n)*a(n-6) + 3*(2*n-7)*a(n-5) + (11-4*n)*a(n-4) + 12*(n-2)*a(n-3) + (5-4*n)*a(n-2) + 3*(2*n-1)*a(n-1), n>=6. - Fung Lam, Feb 19 2014

A162548 A Chebyshev transform of the little Schroeder numbers A001003.

Original entry on oeis.org

1, 1, 2, 9, 37, 156, 695, 3203, 15118, 72739, 355475, 1759624, 8804341, 44457125, 226256114, 1159387253, 5976713665, 30974296468, 161285018771, 843388543471, 4427120165182, 23319497761799, 123221525405447, 652989260163472
Offset: 0

Views

Author

Paul Barry, Jul 05 2009

Keywords

Comments

Hankel transform is Somos-4 variant A162546.

Links

Crossrefs

Cf. A001003, A162543.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1+x+x^2 -Sqrt(1-6*x+3*x^2-6*x^3+x^4))/(4*x*(1+x^2)) )); // G. C. Greubel, Feb 26 2019
  • Mathematica
    CoefficientList[Series[(1+x+x^2 -Sqrt[1-6*x+3*x^2-6*x^3+x^4])/(4*x*(1+x^2)), {x,0,30}], x] (* G. C. Greubel, Feb 26 2019 *)
  • PARI
    my(x='x+O('x^30)); Vec((1+x+x^2 -sqrt(1-6*x+3*x^2-6*x^3+x^4))/( 4*x*(1+x^2))) \\ G. C. Greubel, Feb 26 2019
  • Sage
    ((1+x+x^2 -sqrt(1-6*x+3*x^2-6*x^3+x^4))/(4*x*(1+x^2))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 26 2019

Formula

G.f.: (1/(1+x^2))*s(x/(1+x^2)), s(x) the g.f. of A001003.
G.f.: (1+x+x^2 - sqrt(1-6*x+3*x^2-6*x^3+x^4))/(4*x*(1+x^2)).
G.f.: 1/(1+x^2-x/(1-2*x/(1+x^2-x/(1-2*x/(1+x^2-x/(1-2*x/(1+x+x^2/(1-... (continued fraction);
a(n) = Sum_{k=0..floor(n/2)} (-1)^k*C(n-k,k)*A001003(n-2k).
Conjecture: (n+1)*a(n) +3*(-2*n+1)*a(n-1) +(4*n-5)*a(n-2) +12*(-n+2)*a(n-3) +(4*n-11)*a(n-4) +3*(-2*n+7)*a(n-5) +(n-5)*a(n-6)=0. - R. J. Mathar, Nov 15 2012. (Formula verified and used for computations. - Fung Lam, Feb 19 2014)
Showing 1-4 of 4 results.

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