A180856 Triangle read by rows: T(n,k) is the number of unordered pairs of nodes at distance k in the Moebius ladder M(n) (entries in row n are the coefficients of the corresponding Wiener polynomial). The Moebius ladder M(n) (n>=3) is the graph obtained from the cycle graph C(2n) by adding new edges joining each pair of opposite nodes.

9, 6, 12, 16, 15, 20, 10, 18, 24, 24, 21, 28, 28, 14, 24, 32, 32, 32, 27, 36, 36, 36, 18, 30, 40, 40, 40, 40, 33, 44, 44, 44, 44, 22, 36, 48, 48, 48, 48, 48, 39, 52, 52, 52, 52, 52, 26, 42, 56, 56, 56, 56, 56, 56, 45, 60, 60, 60, 60, 60, 60, 30, 48, 64, 64, 64, 64, 64, 64, 64

Offset

3,1

Comments

Row n contains floor((n+1)/2) entries.

Sum of entries in row n is n(2n-1)=A000384(n).

T(n,1)=3n = number of edges in the corresponding graph.

Sum(k*T(n,k), k>=1) = A180857(n).

References

N. Biggs, Algebraic Graph Theory, 2nd ed. Cambridge University Press, 1993.

Links

Table of n, a(n) for n=3..72.

B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969.

Eric Weisstein's World of Mathematics, Moebius Ladder.

Formula

The generating polynomial for row 2n+1 is n[3t+2t^{(n+1)/2}+4t^2*sum(t^j, j=0..(m-5)/2)] and for row 2n it is n[3t+4t^2*sum(t^j, j=0..(n-4)/2)] (these are also the Wiener polynomials of the corresponding Moebius ladders).

Example

Triangle starts:
9,6;
12,16;
15,20,10;
18,24,24;
21,28,28,14;
24,32,32,32;

Maple

s := proc (m) options operator, arrow: sum(t^j, j = 0 .. m-2) end proc: P := proc (m) if `mod`(m, 2) = 0 then sort(expand(simplify(m*(3*t+4*t^2*s((1/2)*m))))) else sort(expand(simplify(m*(3*t+4*t^2*s((1/2)*m-1/2)+2*t^((1/2)*m+1/2))))) end if end proc: for m from 3 to 16 do P(m) end do: for n from 3 to 16 do seq(coeff(P(n), t, i), i = 1 .. floor((n+1)*1/2)) end do; # yields sequence in triangular form

Crossrefs

Cf. A000384, A180857.

Sequence in context: A063602 A180572 A182873 * A166520 A102648 A272960

Adjacent sequences: A180853 A180854 A180855 * A180857 A180858 A180859

Keyword

nonn,tabf

Author

Emeric Deutsch, Sep 24 2010

Status

approved