A192023 - OEIS

%I #24 Sep 08 2022 08:45:57

%S 1,10,31,68,125,206,315,456,633,850,1111,1420,1781,2198,2675,3216,

%T 3825,4506,5263,6100,7021,8030,9131,10328,11625,13026,14535,16156,

%U 17893,19750,21731,23840,26081,28458,30975,33636,36445,39406,42523,45800,49241,52850,56631

%N 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.

%C The Wiener polynomials of these graphs are given in A192022.

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

%C 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

%H Vincenzo Librandi, <a href="/A192023/b192023.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1)

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

%F G.f.: -x*(-1 - 6*x + 3*x^2) / (x-1)^4. - _R. J. Mathar_, Jun 26 2011

%e 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).

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

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

%Y Cf. A192022.

%K nonn,easy

%O 1,2

%A _Emeric Deutsch_, Jun 24 2011