A187297 Number of 2-step one space leftwards or up, two space rightwards or down asymmetric rook's tours on an n X n board summed over all starting positions.

0, 4, 18, 40, 70, 108, 154, 208, 270, 340, 418, 504, 598, 700, 810, 928, 1054, 1188, 1330, 1480, 1638, 1804, 1978, 2160, 2350, 2548, 2754, 2968, 3190, 3420, 3658, 3904, 4158, 4420, 4690, 4968, 5254, 5548, 5850, 6160, 6478, 6804, 7138, 7480, 7830, 8188, 8554

Offset: 1

Author

R. H. Hardin, Mar 08 2011

Keywords

• nonn
• easy
• changed

Comments

Row 2 of A187296.

For n>=2, a(n) equals the absolute value of 2^n times the x-coefficient of the characteristic polynomial of the n X n matrix with 1/2's along the main diagonal and 1's everywhere else (see Mathematica code below). - John M. Campbell, Jun 21 2011

If (n,2) is an arrangement of n pairs of parallel lines in general position (no two lines from distinct pairs are parallel and no three lines from distinct pairs intersect) then a(n) gives the number of bounded edges in the arrangement. Wetzel and Alexanderson refer to this arrangement as plaid in general position. - Anthony Hernandez, Aug 08 2016

Links

• R. H. Hardin and Vincenzo Librandi, Table of n, a(n) for n = 1..1000 (first 50 terms from R. H. Hardin)
• G. L. Alexanderson and John E. Wetzel, Divisions of Space by Parallels, Transactions of the American Mathematical Society, Volume 291, Number 1 (September 1985), 366-377.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Programs

Mathematica

Table[Abs[ 2^(n)*Coefficient[ CharacteristicPolynomial[ Array[KroneckerDelta[#1, #2]*(1/2 - 1) + 1 &, {n, n}], x], x]], {n, 2, 55}] (* John M. Campbell, Jun 21 2011 *)
Table[If[n == 0, 0, n + n^2 - 2], {n, 0, 200, 2}]  (* Vladimir Joseph Stephan Orlovsky, Jun 26 2011 *)
CoefficientList[Series[2 x (2 + 3 x - x^2)/(1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Feb 08 2014 *)

PARI

a(n)=if(n>1, 4*n^2-6*n, 0) \\ Charles R Greathouse IV, Aug 08 2016

Formula

Empirical: a(n) = 4*n^2 - 6*n = 2*A014107(n) for n>1 (this is now known to be correct - see other comments).

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

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