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.

A091002 Number of walks of length n between non-adjacent nodes on the Petersen graph.

Original entry on oeis.org

0, 0, 1, 2, 9, 22, 77, 210, 673, 1934, 5973, 17578, 53417, 158886, 479389, 1432706, 4309041, 12905278, 38759525, 116191194, 348748345, 1045895510, 3138385581, 9413758642, 28244072129, 84726623982, 254191056757, 762550800650, 2287697141193, 6863001945094
Offset: 0

Views

Author

Paul Barry, Dec 12 2003

Keywords

Comments

Binomial transform of A091005.

Links

Programs

  • GAP
    List([0..30], n -> (3^(n+1) - (-2)^(n+1) - 5)/30); # G. C. Greubel, Feb 01 2019
  • Magma
    [(3^(n+1) - (-2)^(n+1) - 5)/30: n in [0..30]]; // G. C. Greubel, Feb 01 2019
  • Maple
    a:=n->sum(binomial(n-k, k)*6^(k-1), k=1..n): seq(a(n),n=0..27); # Zerinvary Lajos, Sep 30 2006
  • Mathematica
    Table[(3^n -(-2)^n - 5)/30, {n, 40}] (* Vladimir Joseph Stephan Orlovsky, Jun 20 2011 *)
LinearRecurrence[{2,5,-6}, {0,0,1}, 30] (* G. C. Greubel, Feb 01 2019 *)
  • PARI
    vector(30, n, n--; (3^(n+1) - (-2)^(n+1) - 5)/30) \\ G. C. Greubel, Feb 01 2019
  • Sage
    from sage.combinat.sloane_functions import recur_gen2b; it = recur_gen2b(1,2,1,6, lambda n: 1); [next(it) for i in range(0,29)] # Zerinvary Lajos, Jul 03 2008
  • Sage
    [(3^(n+1) - (-2)^(n+1) - 5)/30 for n in range(30)] # G. C. Greubel, Feb 01 2019

Formula

3^n = A091000(n) + 3*A091001(n) + 6*a(n).
G.f.: x^2/((1-x)*(1+2*x)*(1-3*x)).
a(n) = (3^(n+1) - (-2)^(n+1) - 5)/30.
a(n) = (A000244(n) - A001045(n+1)*(-1)^n + 4*A001045(n)*(-1)^n)/10.
a(n) = Sum_{k=1..n} binomial(n-k, k)*6^(k-1). - Zerinvary Lajos, Sep 30 2006
E.g.f.: (3*exp(3*x) + 2*exp(-2*x) - 5*exp(x))/30. - G. C. Greubel, Feb 01 2019

A091004 Expansion of x*(1-x)/((1-2*x)*(1+3*x)).

Original entry on oeis.org

0, 1, -2, 8, -20, 68, -188, 596, -1724, 5300, -15644, 47444, -141308, 425972, -1273820, 3829652, -11472572, 34450484, -103285916, 309988820, -929704316, 2789637236, -8367863132, 25105686548, -75312865340, 225946984628, -677824176668, 2033506084436
Offset: 0

Views

Author

Paul Barry, Dec 13 2003

Keywords

Comments

Inverse binomial transform of A091001.

Links

Crossrefs

Cf. A091001, A091003, A091005, A112555.

Programs

  • GAP
    Concatenation([0], List([1..30], n -> (3*2^n - 8*(-3)^n)/30)); # G. C. Greubel, Feb 01 2019
  • Magma
    [0] cat [(3*2^n - 8*(-3)^n)/30: n in [1..30]]; // G. C. Greubel, Feb 01 2019
  • Mathematica
    CoefficientList[Series[x(1-x)/((1-2x)(1+3x)), {x, 0, 30}], x] (* Wesley Ivan Hurt, Jan 17 2017 *)
Join[{0}, LinearRecurrence[{-1, 6}, {1, -2}, 30]] (* G. C. Greubel, Feb 01 2019 *)
  • PARI
    vector(30, n, n--; (3*2^n - 8*(-3)^n + 5*0^n)/30) \\ G. C. Greubel, Feb 01 2019
  • Sage
    [0] + [(3*2^n - 8*(-3)^n)/30 for n in (1..30)] # G. C. Greubel, Feb 01 2019

Formula

G.f.: x*(1-x)/((1-2*x)*(1+3*x)).
a(n) = (3*2^n - 8*(-3)^n + 5*0^n)/30.
2^n = A091003(n) + 3*a(n) + 6*A091005(n).
a(n+1) = Sum_{k=0..n} A112555(n,k)*(-3)^k. - Philippe Deléham, Sep 11 2009
E.g.f.: (3*exp(2*x) - 8*exp(-3*x) + 5)/30. - G. C. Greubel, Feb 01 2019

A091000 Number of closed walks of length n on the Petersen graph rooted at a given vertex.

Original entry on oeis.org

1, 0, 3, 0, 15, 12, 99, 168, 759, 1764, 6315, 16896, 54783, 156156, 484851, 1421784, 4330887, 12861588, 38846907, 116016432, 349097871, 1045196460, 3139783683, 9410962440, 28249664535, 84715439172, 254213426379, 762506061408
Offset: 0

Views

Author

Paul Barry, Dec 12 2003

Keywords

Comments

If p >= 7 is a prime, then p divides a(p) (provable by easy application of Fermat's Little Theorem). - Adam P. Goucher, Sep 11 2013

Links

Programs

  • GAP
    List([0..30], n -> (3^n+(-2)^(n+2)+5)/10); # G. C. Greubel, Feb 01 2019
  • Magma
    [(3^n+(-2)^(n+2)+5)/10: n in [0..30]]; // G. C. Greubel, Feb 01 2019
  • Mathematica
    Table[{1,0,0}.MatrixPower[{{0,3,0},{1,0,2},{0,1,2}},n].{1,0,0},{n,1,100}] (* Adam P. Goucher, Sep 11 2013 *)
LinearRecurrence[{2,5,-6}, {1,0,3}, 30] (* G. C. Greubel, Feb 01 2019 *)
  • PARI
    vector(30, n, n--; (3^n+(-2)^(n+2)+5)/10) \\ G. C. Greubel, Feb 01 2019
  • Sage
    [(3^n+(-2)^(n+2)+5)/10 for n in (0..30)] # G. C. Greubel, Feb 01 2019

Formula

G.f.: (1-2*x-2*x^2)/((1-x)*(1+2*x)*(1-3*x)).
a(n) = (3^n + (-2)^(n+2) + 5)/10.
a(n) = (A000244(n) + 9*A001045(n+1)(-1)^n + 6*A001045(n)(-1)^(n+1))/10.
3^n = a(n) + 3*A091001(n) + 6*A091002(n)
E.g.f.: (exp(3*x) + 4*exp(-2*x) + 5*exp(x))/10. - G. C. Greubel, Feb 01 2019

A232015 Expansion of (1-2*x)/((1+2*x)*(1-3*x)).

Original entry on oeis.org

1, -1, 5, -1, 29, 23, 197, 335, 1517, 3527, 12629, 33791, 109565, 312311, 969701, 2843567, 8661773, 25723175, 77693813, 232032863, 698195741, 2090392919, 6279567365, 18821924879, 56499329069, 169430878343, 508426852757, 1525012122815, 4575573239357
Offset: 0

Views

Author

Philippe Deléham, Nov 17 2013

Keywords

Links

Crossrefs

Cf. A078008, A108561.

Programs

  • Mathematica
    Table[(3^n + 4 (-2)^n)/5, {n, 0, 30}] (* Bruno Berselli, Nov 18 2013 *)
CoefficientList[Series[(1-2x)/((1+2x)(1-3x)),{x,0,40}],x] (* or *) LinearRecurrence[ {1,6},{1,-1},30] (* Harvey P. Dale, Apr 20 2017 *)
  • PARI
    Vec((1-2*x)/((1+2*x)*(1-3*x))+O(x^20)) \\ Edward Jiang, Sep 06 2014

Formula

G.f.: (1 - 2*x) / (1 - x - 6*x^2).
a(n) = a(n-1) + 6*a(n-2) for n>1, a(0)=1, a(1)=-1.
a(n) = sum_{k=0..n} A108561(n,k)*2^k.
a(n) = A102901(n) - A015441(n).
From Bruno Berselli, Nov 18 2013: (Start)
a(n) = (3^n + 4*(-2)^n)/5.
a(n+1) + a(n) = 4*A015441(n).
a(n+1) - a(n) = -2*(-1)^n*A165405(n).
Sum(a(i), i=0..n) = A091001(n+1). (End)
Showing 1-4 of 4 results.

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