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.

A095364 Number of walks of length n between two adjacent nodes in the cycle graph C_9.

Original entry on oeis.org

1, 0, 3, 0, 10, 0, 35, 1, 126, 11, 462, 78, 1716, 455, 6435, 2380, 24311, 11628, 92398, 54264, 352947, 245157, 1354102, 1081575, 5215250, 4686826, 20156580, 20030039, 78152535, 84672780, 303906051, 354822776, 1184959314, 1476390160
Offset: 1

Views

Author

Herbert Kociemba, Jul 03 2004

Keywords

Comments

In general 2^n/m*Sum(r,0,m-1,Cos(2Pi*k*r/m)Cos(2Pi*r/m)^n) is the number of walks of length n between two nodes at distance k in the cycle graph C_m. Here we have m=9 and k=1.
Also, with offset 2, the cogrowth sequence of the 18-element group D9 = . - Sean A. Irvine, Nov 14 2024

Links

Crossrefs

Cf. A095367, A095368, A095369.
Cf. A007582 (D8), A377573 (D7).

Programs

  • PARI
    a(n) = round(2^n/9*sum(r=0, 8, cos(2*Pi*r/9)^(n+1))) \\ Michel Marcus, Jul 18 2013
  • PARI
    Vec( x*(-1+x+2*x^2-x^3)/((1+x)*(-1+2*x)*(1-3*x^2+x^3))+O(x^66) ) \\ Joerg Arndt, Jul 18 2013

Formula

a(n) = 2^n/9 * sum(r=0..8, cos(2*Pi*r/9)^(n+1)).
G.f.: x(-1+x+2x^2-x^3)/((1+x)(-1+2x)(1-3x^2+x^3)).
a(n) = a(n-1) + 5*a(n-2) - 4*a(n-3) - 5*a(n-4) + 2*a(n-5).

A095367 Number of walks of length n between two nodes at distance 2 in the cycle graph C_9.

Original entry on oeis.org

1, 0, 4, 0, 15, 1, 56, 9, 210, 56, 792, 299, 3003, 1470, 11441, 6868, 43776, 31008, 168151, 136629, 648208, 591261, 2507046, 2523676, 9726080, 10656387, 37839375, 44612702, 147600981, 185477216, 577147212, 766744608, 2261792303
Offset: 2

Views

Author

Herbert Kociemba, Jul 03 2004

Keywords

Comments

In general (2^n/m)*Sum_{r=0..m-1} cos(2*Pi*k*r/m)*cos(2*Pi*r/m)^n is the number of walks of length n between two nodes at distance k in the cycle graph C_m. Here we have m=9 and k=2.

Links

Crossrefs

Cf. A095364, A095368, A095369.

Formula

a(n) = (2^n/9)*Sum_{r=0..8} cos(4*Pi*r/9)*cos(2*Pi*r/9)^n.
G.f.: x^2(-1+x+x^2)/((1+x)*(-1+2x)*(1-3x^2+x^3));
a(n) = a(n-1) + 5*a(n-2) - 4*a(n-3) - 5*a(n-4) + 2*a(n-5).

A095368 Number of walks of length n between two nodes at distance 3 in the cycle graph C_9.

Original entry on oeis.org

1, 0, 5, 1, 21, 8, 84, 45, 330, 221, 1287, 1015, 5006, 4488, 19465, 19380, 75753, 82365, 295261, 346104, 1152944, 1442101, 4510830, 5969561, 17682795, 24582663, 69448446, 100804436, 273241161, 411921832, 1076832989
Offset: 3

Views

Author

Herbert Kociemba, Jul 03 2004

Keywords

Comments

In general (2^n/m)*Sum_{r=0..m-1} cos(2Pi*k*r/m)*cos(2Pi*r/m)^n is the number of walks of length n between two nodes at distance k in the cycle graph C_m. Here we have m=9 and k=3.

Links

Crossrefs

Cf. A095364, A095367, A095369.

Programs

  • Mathematica
    LinearRecurrence[{1,5,-4,-5,2},{1,0,5,1,21},40] (* Harvey P. Dale, Dec 21 2024 *)

Formula

a(n) = (2^n/9)*Sum_{r=0..8} cos(2Pi*r/3)*cos(2Pi*r/9)^n.
G.f.: (-1+x)x^3/((1+x)(-1+2x)(1-3x^2+x^3)).
a(n) = a(n-1)+5a(n-2)-4a(n-3)-5a(n-4)+2a(n-5).
Showing 1-3 of 3 results.

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

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