A394278 - OEIS

%I #6 Mar 14 2026 17:16:00

%S 1,4,9,16,33,73,129,205,331,529,779,1097,1531,2075,2716,3475,4376,

%T 5399,6544,7824,9232,10768,12432,14224,16144,18192,20368,22672,25104,

%U 27664,30352,33168,36112,39184,42384,45712,49168,52752,56464,60304,64272,68368,72592

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

%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/FiveleaperGraph.html">Fiveleaper Graph</a>.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F a(n) = 16*(409 - 76*n + 4*n^2) for n > 18.

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

%F G.f.: x*(-1-x-8*x^4-13*x^5+7*x^6-4*x^7-30*x^8-22*x^9+20*x^10-16*x^11-48*x^12+6*x^13+13*x^14-21*x^15-24*x^16+20*x^17-13*x^19+7*x^20)/(-1+x)^3.

%t Table[n^2 + Ramp[n - 19]^2 + 4 Ramp[n - 16] Ramp[n - 15] + 2 Ramp[n - 19] Ramp[n - 14] + Ramp[n - 14]^2 + 4 Ramp[n - 12]^2 + 4 Ramp[n - 15] Ramp[n - 11] + 2 Ramp[n - 13] Ramp[n - 11] + 4 Ramp[n - 16] Ramp[n - 10] + 4 Ramp[n - 11] Ramp[n - 10] + 2 Ramp[n - 11] Ramp[n - 8] + 4 Ramp[n - 9] Ramp[n - 8] + 8 Ramp[n - 12] Ramp[n - 7] + 4 Ramp[n - 7]^2 + 2 Ramp[n - 13] Ramp[n - 6] + 2 Ramp[n - 8] Ramp[n - 6] + 2 n Ramp[n - 5] + Ramp[n - 5]^2 + 4 Ramp[n - 8] Ramp[n - 4] + 4 Ramp[n - 9] Ramp[n - 3] + 4 Ramp[n - 4] Ramp[n - 3], {n, 20}]

%t Join[{1, 4, 9, 16, 33, 73, 129, 205, 331, 529, 779, 1097, 1531, 2075, 2716, 3475, 4376, 5399}, LinearRecurrence[{3, -3, 1}, {6544, 7824, 9232}, 2]]

%t CoefficientList[Series[(-1 - x - 8 x^4 - 13 x^5 + 7 x^6 - 4 x^7 - 30 x^8 - 22 x^9 + 20 x^10 - 16 x^11 - 48 x^12 + 6 x^13 + 13 x^14 - 21 x^15 - 24 x^16 + 20 x^17 - 13 x^19 + 7 x^20)/(-1 + x)^3, {x, 0, 20}], x]

%K nonn,easy

%O 1,2

%A _Eric W. Weisstein_, Mar 14 2026