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Wiener index of the n X n black bishop graph.
0

%I #8 Sep 08 2017 09:12:06

%S 0,1,14,42,124,251,506,852,1432,2165,3270,4606,6484,8687,11634,15016,

%T 19376,24297,30462,37330,45740,55011,66154,78332,92744,108381,126646,

%U 146342,169092,193495,221410,251216,285024,320977,361454,404346,452316,502987,559322,618660

%N Wiener index of the n X n black bishop graph.

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

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

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-6,0,6,-2,-2,1).

%F a(n) = (3 - 10*n + 6*n^2 - 8*n^3 + 6*n^4 - 3*(-1)^n*(1 - 2*n + 2*n^2))/24.

%F a(n) = 2*a(n-1) + 2*a(n-2) - 6*a(n-3) + 6*a(n-5) - 2*a(n-6) - 2*a(n-7) + a(n-8).

%F G.f.: x^2*(-1 - 12*x - 12*x^2 - 18*x^3 - 3*x^4 - 2*x^5)/((-1 + x)^5*(1 + x)^3).

%t Table[(3 - 10 n + 6 n^2 - 8 n^3 + 6 n^4 - 3 (-1)^n (1 - 2 n + 2 n^2))/24, {n, 20}]

%t LinearRecurrence[{2, 2, -6, 0, 6, -2, -2, 1}, {0, 1, 14, 42, 124, 251, 506, 852}, 20]

%t CoefficientList[Series[x (-1 - 12 x - 12 x^2 - 18 x^3 - 3 x^4 - 2 x^5)/((-1 + x)^5 (1 + x)^3), {x, 0, 20}], x]

%K nonn,easy

%O 1,3

%A _Eric W. Weisstein_, Sep 08 2017