A394293 - OEIS

%I #4 Mar 15 2026 10:11:41

%S 0,3,9,25,51,98,166,270,410,605,855,1183,1589,2100,2716,3468,4356,

%T 5415,6645,8085,9735,11638,13794,16250,19006,22113,25571,29435,33705,

%U 38440,43640,49368,55624,62475,69921,78033,86811,96330,106590,117670,129570,142373

%N Number of induced cubes in the n X n white bishop graph.

%C Sequence extended to a(1) using the formula.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CubePolynomial.html">Cube Polynomial</a>.

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

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-5,5,1,-3,1).

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

%F a(n) = 3*a(n-1)-a(n-2)-5*a(n-3)+5*a(n-4)+a(n-5)-3*a(n-6)+a(n-7).

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

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

%t Table[Piecewise[{{n (n + 1)^2 (n + 2)/24, Mod[n, 2] == 0}, {(n - 1) (n + 1) (n^2 + 4 n + 6)/24, Mod[n, 2] == 1}}], {n, 20}]

%t LinearRecurrence[{3, -1, -5, 5, 1, -3, 1}, {3, 9, 25, 51, 98, 166, 270}, 20]

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

%K nonn

%O 1,2

%A _Eric W. Weisstein_, Mar 15 2026