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

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

A102765 Array read by antidiagonals: T(n, k) = ((n+4)^k-(n-1)^k)/5.

Original entry on oeis.org

0, 0, 1, 0, 1, 3, 0, 1, 5, 13, 0, 1, 7, 25, 51, 0, 1, 9, 43, 125, 205, 0, 1, 11, 67, 259, 625, 819, 0, 1, 13, 97, 477, 1555, 3125, 3277, 0, 1, 15, 133, 803, 3355, 9331, 15625, 13107, 0, 1, 17, 175, 1261, 6505, 23517, 55987, 78125, 52429, 0, 1, 19, 223, 1875, 11605
Offset: 0

Views

Author

Lambert Klasen (lambert.klasen(AT)gmx.net) and Gary W. Adamson, Feb 10 2005

Keywords

Comments

Consider a 5x5 matrix M =
[n, 1, 1, 1, 1]
[1, n, 1, 1, 1]
[1, 1, n, 1, 1]
[1, 1, 1, n, 1]
[1, 1, 1, 1, n].
The n-th row of the array contains the values of the non diagonal elements of M^k, k=0,1,.... (Corresponding diagonal entry = non diagonal entry + (n-1)^k.)
For row r we have polynomial ((r+4)^n-(r-1)^n)/5. Corresponding g.f.s: x/((1-(r-1)x)(1-(r+4)x))
If r(n) denotes a row sequence, r(n+1)/r(n) converges to n+4.
Triangle T(n, k) = (4^(n-k-1)-(-1)^(n-k-1))/5*(binomial(k+(n-k-1),n-k-1)) gives coefficients for polynomials for the columns of the array. First four polynomial are:
1
3 + 2*k
13 + 9*k + 3*k^2
51 + 52*k + 18*k^2 + 4*k^3
...

Examples

			Array begins:
  0, 1,  3, 13,  51,  205, ...
  0, 1,  5, 25, 125,  625, ...
  0, 1,  7, 43, 259, 1555, ...
  0, 1,  9, 67, 477, 3355, ...
  0, 1, 11, 97, 803, 6505, ...
  ...

Crossrefs

Cf. A015521 (for n=0), A000351 (for n=1), A003464 (for n=2), A016130 (for n=3), A016140 (for n=4), A016153 (for n=5), A016164 (for n=6), A016174 (for n=7), A016184 (for n=8), A015441 (for n=-1), A091005 (for n=-2).

Programs

  • PARI
    MM(n,N)=local(M);M=matrix(n,n);for(i=1,n, for(j=1,n,if(i==j,M[i,j]=N,M[i,j]=1)));M
for(k=0,10, for(i=0,10,print1((MM(5,k)^i)[1,2],","));print())
  • PARI
    p(n,k)=((n+4)^k-(n-1)^k)/5
for(k=0,10, for(i=0,10,print1(p(k,i),","));print())
  • PARI
    for(k=0,10, for(i=0,10,print1(polcoeff(x/((1-(k-1)*x)*(1-(k+4)*x)),i),","));print())
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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