%I #24 Oct 07 2023 16:06:41

%S 0,0,8,34,105,248,490,858,1379,2080,2988,4130,5533,7224,9230,11578,

%T 14295,17408,20944,24930,29393,34360,39858,45914,52555,59808,67700,

%U 76258,85509,95480,106198,117690,129983,143104,157080,171938,187705

%N Number of ways to place 3 nonattacking kings on a 3 X n board.

%H Vincenzo Librandi, <a href="/A172202/b172202.txt">Table of n, a(n) for n = 1..1000</a>

%H Vaclav Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Number of ways of placing non-attacking queens and kings on boards of various sizes</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-2)*(9*n^2 - 45*n + 70)/2, n>=2.

%F G.f.: x^3*(8+2*x+17*x^2)/(1-x)^4. - _Vaclav Kotesovec_, Mar 24 2010

%F E.g.f.: 70 + 17*x + (1/2)*(-140 + 106*x - 36*x^2 + 9*x^3)*exp(x). - _G. C. Greubel_, Apr 29 2022

%t CoefficientList[Series[x^2*(8+2*x+17*x^2)/(1-x)^4, {x, 0, 50}], x] (* _Vincenzo Librandi_, May 27 2013 *)

%t LinearRecurrence[{4,-6,4,-1},{0,0,8,34,105},40] (* _Harvey P. Dale_, Oct 07 2023 *)

%o (Magma) [0] cat [(n-2)*(9*n^2-45*n+70)/2: n in [2..50]]; // _G. C. Greubel_, Apr 29 2022

%o (SageMath) [(1/8)*(n-2)*(9*(2*n-5)^2+55) +17*bool(n==1) for n in (1..50)] # _G. C. Greubel_, Apr 29 2022

%Y Cf. A047659, A061989, A061996.

%K nonn,easy

%O 1,3

%A _Vaclav Kotesovec_, Jan 29 2010

%E More terms from _Vincenzo Librandi_, May 27 2013