A172202 Number of ways to place 3 nonattacking kings on a 3 X n board.
0, 0, 8, 34, 105, 248, 490, 858, 1379, 2080, 2988, 4130, 5533, 7224, 9230, 11578, 14295, 17408, 20944, 24930, 29393, 34360, 39858, 45914, 52555, 59808, 67700, 76258, 85509, 95480, 106198, 117690, 129983, 143104, 157080, 171938, 187705
Offset: 1
Links
- Vincenzo Librandi, Table of n, a(n) for n = 1..1000
- Vaclav Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes
- Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
Programs
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Magma
[0] cat [(n-2)*(9*n^2-45*n+70)/2: n in [2..50]]; // G. C. Greubel, Apr 29 2022
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Mathematica
CoefficientList[Series[x^2*(8+2*x+17*x^2)/(1-x)^4, {x, 0, 50}], x] (* Vincenzo Librandi, May 27 2013 *) LinearRecurrence[{4,-6,4,-1},{0,0,8,34,105},40] (* Harvey P. Dale, Oct 07 2023 *) -
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
Formula
a(n) = (n-2)*(9*n^2 - 45*n + 70)/2, n>=2.
G.f.: x^3*(8+2*x+17*x^2)/(1-x)^4. - Vaclav Kotesovec, Mar 24 2010
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
Extensions
More terms from Vincenzo Librandi, May 27 2013