A180858 - OEIS

%I #6 Mar 30 2020 04:18:00

%S 1,4,3,9,7,7,16,13,12,13,25,21,19,19,21,36,31,28,27,28,31,49,43,39,37,

%T 37,39,43,64,57,52,49,48,49,52,57,81,73,67,63,61,61,63,67,73,100,91,

%U 84,79,76,75,76,79,84,91,121,111,103,97,93,91,91,93,97,103,111,144,133,124

%N Square array read by antidiagonals: T(p,n) is the Wiener index of the fan graph F(p,n) (p>=1, n>=1). F(p,n) is the graph obtained by placing an edge between each node of the empty graph on p nodes and each node of the path graph on n nodes.

%C The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices in the graph.

%H B. E. Sagan, Y-N. Yeh and P. Zhang, <a href="http://users.math.msu.edu/users/sagan/Papers/Old/wpg-pub.pdf">The Wiener Polynomial of a Graph</a>, Internat. J. of Quantum Chem., 60, 1996, 959-969.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FanGraph.html">Fan Graph</a>.

%F T(p,n) = p(p-1)+(n-1)^2+pn.

%F T(1,n) = n^2-n+1 = A002061(n).

%F T(2,n) = n^2+3 = A117950(n).

%F The Wiener polynomial of the graph F(p,n) is (pn+n-1)t+(1/2)[p(p-1)+(n-1)(n-2)]t^2.

%e T(2,3)=12 because the corresponding fan graph is the wheel graph on 5 nodes OABCD, O being the center of the wheel. Its Wiener index = number of edges + |AC| +|BD| = 8+2+2=12.

%e Square array T(p,n) begins:

%e 1, 3, 7, 13, 21, 31, 43,...

%e 4, 7, 12, 19, 28, 39, 52, ...

%e 9, 13, 19, 27, 37, 49, 63,...

%e 16, 21, 28, 37, 48, 61, 76, ...

%p T := proc (p, n) options operator, arrow: p*(p-1)+(n-1)^2+p*n end proc: for i to 12 do seq(T(i+1-j, j), j = 1 .. i) end do; # yields sequence in triangular form

%Y Cf. A002061, A117950

%K nonn,tabl

%O 1,2

%A _Emeric Deutsch_, Sep 24 2010