A192023 The Wiener index of the comb-shaped graph |||...|_| with 2n (n>=1) nodes. The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices in the graph.

1, 10, 31, 68, 125, 206, 315, 456, 633, 850, 1111, 1420, 1781, 2198, 2675, 3216, 3825, 4506, 5263, 6100, 7021, 8030, 9131, 10328, 11625, 13026, 14535, 16156, 17893, 19750, 21731, 23840, 26081, 28458, 30975, 33636, 36445, 39406, 42523, 45800, 49241, 52850, 56631

Offset: 1

Author

Emeric Deutsch, Jun 24 2011

Keywords

• nonn
• easy

Comments

The Wiener polynomials of these graphs are given in A192022.

a(n) = Sum_{k>=1} A192022(n,k).

Conjecture: for n>2, A192023(n-2) is the number of 2 X 2 matrices with all terms in {1,2,...,n} and determinant 2n. - Clark Kimberling, Mar 31 2012

Examples

			a(2)=10 because in the graph |_| there are 3 pairs of nodes at distance 1, 2 pairs at distance 2, and 1 pair at distance 3 (3*1 + 2*2 + 1*3 = 10).
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1)

Crossrefs

Cf. A192022.

Programs

Magma

[n*(2*n^2+6*n-5)/3: n in [1..50]]; // Vincenzo Librandi, Jul 04 2011

Maple

a := proc (n) options operator: arrow: (1/3)*n*(2*n^2+6*n-5) end proc: seq(a(n), n = 1 .. 43);

Formula

a(n) = n*(2*n^2 + 6*n - 5)/3.

G.f.: -x*(-1 - 6*x + 3*x^2) / (x-1)^4. - R. J. Mathar, Jun 26 2011