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

A091001 Number of walks of length n between adjacent nodes on the Petersen graph.

Original entry on oeis.org

0, 1, 0, 5, 4, 33, 56, 253, 588, 2105, 5632, 18261, 52052, 161617, 473928, 1443629, 4287196, 12948969, 38672144, 116365957, 348398820, 1046594561, 3136987480, 9416554845, 28238479724, 84737808793, 254168687136, 762595539893
Offset: 0

Views

Author

Paul Barry, Dec 12 2003

Keywords

References

  • N. Biggs, Algebraic Graph Theory, Cambridge, 2nd. Ed., 1993, p. 20.
  • F. Harary, Graph Theory, Addison-Wesley, 1969, p. 89.

Links

Programs

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

Formula

G.f.: x*(1-2*x)/((1-x)*(1+2*x)*(1-3*x)).
a(n) = (3^(n+1) + (-2)^(n+3) + 5)/30.
3^n = A091000(n) + 3*a(n) + 6*A091002(n).
a(n) = (A000244(n) - A001045(n+1)*(-1)^n - 6*A001045(n)*(-1)^n)/10.
a(n) = A091002(n+1) - 2*A091002(n). - R. J. Mathar, Oct 30 2014
E.g.f.: (3*exp(3*x) - 8*exp(-2*x) +5*exp(x))/30. - G. C. Greubel, Feb 01 2019

A091003 Expansion of (1-3*x^2)/((1-2*x)*(1+3*x)).

Original entry on oeis.org

1, -1, 4, -10, 34, -94, 298, -862, 2650, -7822, 23722, -70654, 212986, -636910, 1914826, -5736286, 17225242, -51642958, 154994410, -464852158, 1394818618, -4183931566, 12552843274, -37656432670, 112973492314, -338912088334, 1016753042218
Offset: 0

Views

Author

Paul Barry, Dec 13 2003

Keywords

Comments

Inverse binomial transform of A091000.

Links

Programs

  • GAP
    Concatenation([1], List([1..30], n -> (2^n + 4*(-3)^n)/10)); # G. C. Greubel, Feb 01 2019
  • Magma
    [1] cat [(2^n + 4*(-3)^n)/10: n in [1..30]]; // G. C. Greubel, Feb 01 2019
  • Mathematica
    CoefficientList[Series[(1-3x^2)/((1-2x)(1+3x)),{x,0,30}],x] (* Harvey P. Dale, Dec 23 2014 *)
Join[{1}, LinearRecurrence[{-1,6}, {-1,4}, 30]] (* G. C. Greubel, Feb 01 2019 *)
  • PARI
    vector(30, n, n--; (2^n + 4*(-3)^n + 5*0^n)/10) \\ G. C. Greubel, Feb 01 2019
  • Sage
    [1] + [(2^n + 4*(-3)^n)/10 for n in (1..30)] # G. C. Greubel, Feb 01 2019

Formula

2^n = A091003(n) + 3*A091004(n) + 6*A091005(n).
a(n) = (2^n + 4*(-3)^n + 5*0^n)/10.
E.g.f.: (exp(2*x) + 4*exp(-3*x) + 5)/10. - G. C. Greubel, Feb 01 2019

A229031 Number of 5-colorings of the strong product of the complete graph K2 and the cycle graph Cn.

Original entry on oeis.org

120, 0, 2400, 3840, 63360, 215040, 1943040, 9031680, 64665600, 346030080, 2243911680, 12792299520, 79437987840, 465890181120, 2838290104320, 16857940623360, 101834835886080, 608260231004160, 3660556491816960, 21919358464819200, 131692072607416320, 789448748118835200, 4739507238312345600, 28425784430470103040
Offset: 2

Views

Author

Adam P. Goucher, Sep 11 2013

Keywords

Comments

The strong product of K2 and Cn can be regarded as the King's graph on a 2*n cylindrical (or equivalently toroidal) chessboard.
The Kneser graph construction of the Petersen graph relates this to the number of closed walks on the Petersen graph.
More generally, the number of c-colorings of the strong product of Km and Cn is equal to (m!)^n * (c choose m) * (number of closed walks of length n on K(c,m)).
If n is prime then a(n) is divisible by n, since the cyclic group of order n acts on the colorings, partitioning them into orbits of size n. More generally, n divides a(n) for any Carmichael number n, due to the closed form.

Examples

			For n = 2, the graph is the complete graph K4, which has a(4) = 120 different 5-colorings corresponding to ordered 4-subsets of {1,2,3,4,5}.
For n = 3, the graph is the complete graph K6, which cannot be 5-colored, so a(3) = 0. Equivalently, there are no closed walks of length 3 on the Petersen graph.

Links

Programs

  • Mathematica
    Table[2^n(3^n+4(-2)^n+5),{n,2,25}]
LinearRecurrence[{4,20,-48},{120,0,2400},24] (* or *) Drop[CoefficientList[Series[-120*x^2*(4*x - 1) / ((2*x - 1) * (4*x + 1) * (6*x - 1)), {x, 0, 25}], x], 2] (* Indranil Ghosh, Mar 03 2017 *)
  • PARI
    a(n) = (2^n) * (3^n + 4*(-2)^n + 5) \\ Indranil Ghosh, Mar 03 2017
  • Python
    def A229031(n) : return (2**n) * (3**n + 4*(-2)**n +5) # Indranil Ghosh, Mar 03 2017

Formula

a(n) = 6^n + 4*(-4)^n + 5*2^n.
a(n) = 10 * 2^n * A091000(n).
a(n) = 4*a(n-1)+20*a(n-2)-48*a(n-3). G.f.: -120*x^2*(4*x-1) / ((2*x-1)*(4*x+1)*(6*x-1)). - Colin Barker, Oct 20 2013
Showing 1-4 of 4 results.

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