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A143941 The Wiener index of a chain of n triangles (i.e., joined like VVV..VV; here V is a triangle!).

%I A143941 #51 Feb 16 2025 08:33:08
%S A143941 3,14,37,76,135,218,329,472,651,870,1133,1444,1807,2226,2705,3248,
%T A143941 3859,4542,5301,6140,7063,8074,9177,10376,11675,13078,14589,16212,
%U A143941 17951,19810,21793,23904,26147,28526,31045,33708,36519,39482,42601,45880,49323,52934
%N A143941 The Wiener index of a chain of n triangles (i.e., joined like VVV..VV; here V is a triangle!).
%C A143941 The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices in the graph.
%C A143941 Row 2 of the convolution array A213752. - _Clark Kimberling_, Jun 20 2012
%C A143941 Also the circuit rank of the (n+2) X (n+2) bishop graph. - _Eric W. Weisstein_, May 10 2019
%H A143941 Vincenzo Librandi, <a href="/A143941/b143941.txt">Table of n, a(n) for n = 1..1000</a>
%H A143941 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BishopGraph.html">Bishop Graph</a>
%H A143941 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CircuitRank.html">Circuit Rank</a>
%H A143941 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A143941 a(n) = n*(1 + 6*n + 2*n^2)/3.
%F A143941 G.f.: z*(3 + 2*z - z^2)/(1-z)^4.
%F A143941 a(n) = Sum_{k=1..n} k*A143940(n,k).
%F A143941 a(n) = Sum_{k=1..n} A142463(k). - _Richard R. Forberg_, Jan 09 2015
%F A143941 a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - _Wesley Ivan Hurt_, Apr 08 2015
%F A143941 E.g.f.: exp(x)*x*(9 + 12*x + 2*x^2)/3. - _Stefano Spezia_, Jan 03 2022
%e A143941 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 A143941 seq((1/3)*n*(1+6*n+2*n^2), n=1..43);
%t A143941 CoefficientList[Series[(3+2*x-x^2)/(1-x)^4,{x,0,40}],x] (* _Vincenzo Librandi_, Jul 03 2012 *)
%t A143941 LinearRecurrence[{4,-6,4,-1},{3,14,37,76},50] (* _Harvey P. Dale_, Sep 06 2023 *)
%o A143941 (Magma) [n*(1+6*n+2*n^2)/3 : n in [1..40]]; // _Wesley Ivan Hurt_, Apr 08 2015
%Y A143941 Cf. A142463, A143940, A213752.
%K A143941 nonn,easy
%O A143941 1,1
%A A143941 _Emeric Deutsch_, Sep 06 2008