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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. 3
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 (list; graph; refs; listen; history; text; internal format)
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 A283846

Adjacent sequences:  A192020 A192021 A192022 * A192024 A192025 A192026

KEYWORD

nonn,easy

AUTHOR

Emeric Deutsch, Jun 24 2011

STATUS

approved

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Last modified August 9 05:12 EDT 2020. Contains 336319 sequences. (Running on oeis4.)