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

1,2

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

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)

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

Example

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).

Maple

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

Prog

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

Crossrefs

Cf. A192022.

Sequence in context: A085473 A051943 A059306 * A219693 A297507 A163655

Adjacent sequences: A192020 A192021 A192022 * A192024 A192025 A192026

Keyword

nonn,easy

Author

Emeric Deutsch, Jun 24 2011

Status

approved