%I #50 Sep 06 2023 15:58:59

%S 3,14,37,76,135,218,329,472,651,870,1133,1444,1807,2226,2705,3248,

%T 3859,4542,5301,6140,7063,8074,9177,10376,11675,13078,14589,16212,

%U 17951,19810,21793,23904,26147,28526,31045,33708,36519,39482,42601,45880,49323,52934

%N The Wiener index of a chain of n triangles (i.e., joined like VVV..VV; here V is a triangle!).

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

%C Row 2 of the convolution array A213752. - _Clark Kimberling_, Jun 20 2012

%C Also the circuit rank of the (n+2) X (n+2) bishop graph. - _Eric W. Weisstein_, May 10 2019

%H Vincenzo Librandi, <a href="/A143941/b143941.txt">Table of n, a(n) for n = 1..1000</a>

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

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CircuitRank.html">Circuit Rank</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*(1 + 6*n + 2*n^2)/3.

%F G.f.: z*(3 + 2*z - z^2)/(1-z)^4.

%F a(n) = Sum_{k=1..n} k*A143940(n,k).

%F a(n) = Sum_{k=1..n} A142463(k). - _Richard R. Forberg_, Jan 09 2015

%F a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - _Wesley Ivan Hurt_, Apr 08 2015

%F E.g.f.: exp(x)*x*(9 + 12*x + 2*x^2)/3. - _Stefano Spezia_, Jan 03 2022

%e a(2)=14 because in the graph VV (V is a triangle!) we have 6 distances equal to 1 and 4 distances equal to 2.

%p seq((1/3)*n*(1+6*n+2*n^2), n=1..43);

%t CoefficientList[Series[(3+2*x-x^2)/(1-x)^4,{x,0,40}],x] (* _Vincenzo Librandi_, Jul 03 2012 *)

%t LinearRecurrence[{4,-6,4,-1},{3,14,37,76},50] (* _Harvey P. Dale_, Sep 06 2023 *)

%o (Magma) [n*(1+6*n+2*n^2)/3 : n in [1..40]]; // _Wesley Ivan Hurt_, Apr 08 2015

%Y Cf. A142463, A143940, A213752.

%K nonn,easy

%O 1,1

%A _Emeric Deutsch_, Sep 06 2008