A187296 -id:A187296 - cp's OEIS frontend

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

R. H. Hardin, Mar 08 2011

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

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).

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 *)

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

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.

A187298 Number of 3-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.

Original entry on oeis.org

0, 2, 36, 98, 198, 330, 494, 690, 918, 1178, 1470, 1794, 2150, 2538, 2958, 3410, 3894, 4410, 4958, 5538, 6150, 6794, 7470, 8178, 8918, 9690, 10494, 11330, 12198, 13098, 14030, 14994, 15990, 17018, 18078, 19170, 20294, 21450, 22638, 23858, 25110, 26394

Offset: 1

R. H. Hardin, Mar 08 2011

nonn

Row 3 of A187296.

			Some solutions for 4 X 4:
..0..1..0..0....0..0..0..0....0..0..0..0....0..0..1..0....0..0..0..3
..0..0..0..0....1..0..0..0....0..1..0..0....0..0..3..0....0..1..0..2
..3..2..0..0....0..0..0..0....0..3..0..0....0..0..2..0....0..0..0..0
..0..0..0..0....2..0..3..0....0..2..0..0....0..0..0..0....0..0..0..0

R. H. Hardin, Table of n, a(n) for n = 1..50

Cf. A187296.

Empirical: a(n) = 16*n^2 - 44*n + 18 for n>3.
Conjectures from Colin Barker, Apr 23 2018: (Start)
G.f.: 2*x^2*(1 + 15*x - 2*x^2 + 5*x^3 - 3*x^4) / (1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>6.
(End)

A187299 Number of 4-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.

Original entry on oeis.org

0, 0, 54, 196, 480, 876, 1398, 2036, 2790, 3660, 4646, 5748, 6966, 8300, 9750, 11316, 12998, 14796, 16710, 18740, 20886, 23148, 25526, 28020, 30630, 33356, 36198, 39156, 42230, 45420, 48726, 52148, 55686, 59340, 63110, 66996, 70998, 75116, 79350, 83700

Offset: 1

R. H. Hardin, Mar 08 2011

nonn

Row 4 of A187296.

			Some solutions for 4 X 4:
..1..0..0..0....0..0..0..0....0..0..1..0....0..0..0..0....0..0..0..0
..0..0..0..0....0..0..0..0....0..0..0..0....0..2..1..0....2..4..3..0
..2..4..3..0....1..3..2..4....4..3..2..0....0..4..0..0....1..0..0..0
..0..0..0..0....0..0..0..0....0..0..0..0....0..3..0..0....0..0..0..0

R. H. Hardin, Table of n, a(n) for n = 1..50

Cf. A187296.

Empirical: a(n) = 58*n^2 - 232*n + 180 for n>5.
Conjectures from Colin Barker, Apr 23 2018: (Start)
G.f.: 2*x^3*(27 + 17*x + 27*x^2 - 15*x^3 + 7*x^4 - 5*x^5) / (1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>8.
(End)

A187300 Number of 5-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.

Original entry on oeis.org

0, 0, 90, 416, 1208, 2400, 4092, 6208, 8766, 11752, 15166, 19008, 23278, 27976, 33102, 38656, 44638, 51048, 57886, 65152, 72846, 80968, 89518, 98496, 107902, 117736, 127998, 138688, 149806, 161352, 173326, 185728, 198558, 211816, 225502, 239616

Offset: 1

R. H. Hardin, Mar 08 2011

nonn

Row 5 of A187296.

			Some solutions for 4 X 4:
..0..0..0..0....0..4..3..2....0..0..0..0....0..0..0..0....0..0..0..0
..0..0..2..1....0..0..0..1....5..4..3..0....5..1..0..0....3..0..4..0
..0..0..0..0....0..5..0..0....0..0..2..0....4..0..0..0....2..0..0..0
..0..4..3..5....0..0..0..0....0..0..1..0....3..2..0..0....1..0..5..0

R. H. Hardin, Table of n, a(n) for n = 1..50

Cf. A187296.

Empirical: a(n) = 214*n^2 - 1080*n + 1152 for n>7.
Conjectures from Colin Barker, Apr 23 2018: (Start)
G.f.: 2*x^3*(45 + 73*x + 115*x^2 - 33*x^3 + 50*x^4 - 38*x^5 + 9*x^6 - 7*x^7) / (1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>10.
(End)

A187301 Number of 6-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.

Original entry on oeis.org

0, 0, 144, 884, 3006, 6520, 11896, 18832, 27478, 37696, 49508, 62896, 77860, 94400, 112516, 132208, 153476, 176320, 200740, 226736, 254308, 283456, 314180, 346480, 380356, 415808, 452836, 491440, 531620, 573376, 616708, 661616, 708100, 756160

Offset: 1

R. H. Hardin, Mar 08 2011

nonn

Row 6 of A187296.

			Some solutions for 4 X 4:
..0..2..1..3....0..2..1..3....0..5..0..6....0..1..0..0....6..5..0..0
..0..0..0..5....0..0..6..5....2..4..3..0....0..0..5..0....0..4..0..0
..0..0..0..4....0..0..0..4....1..0..0..0....0..2..4..3....1..3..2..0
..0..0..0..6....0..0..0..0....0..0..0..0....0..0..6..0....0..0..0..0

R. H. Hardin, Table of n, a(n) for n = 1..50

Cf. A187296.

Empirical: a(n) = 788*n^2 - 4736*n + 6256 for n>9.
Conjectures from Colin Barker, Apr 23 2018: (Start)
G.f.: 2*x^3*(72 + 226*x + 393*x^2 + 5*x^3 + 235*x^4 - 151*x^5 + 75*x^6 - 69*x^7 + 11*x^8 - 9*x^9) / (1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>12.
(End)

A187302 Number of 7-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.

Original entry on oeis.org

0, 0, 108, 1368, 6264, 15596, 31172, 52256, 79634, 112568, 151266, 195512, 245332, 300704, 361628, 428104, 500132, 577712, 660844, 749528, 843764, 943552, 1048892, 1159784, 1276228, 1398224, 1525772, 1658872, 1797524, 1941728, 2091484, 2246792

Offset: 1

R. H. Hardin, Mar 08 2011

nonn

Row 7 of A187296.

			Some solutions for 4 X 4:
..0..0..0..0....0..0..2..1....0..0..0..0....0..2..1..3....0..0..0..0
..5..4..6..0....0..5..7..6....0..4..3..5....0..7..0..0....7..0..0..1
..0..3..2..1....0..4..3..0....1..0..2..0....0..6..5..4....6..5..0..0
..0..0..7..0....0..0..0..0....0..0..7..6....0..0..0..0....0..4..3..2

R. H. Hardin, Table of n, a(n) for n = 1..50

Cf. A187296.

Empirical: a(n) = 2776*n^2 - 19580*n + 30728 for n>11.
Conjectures from Colin Barker, Apr 23 2018: (Start)
G.f.: 2*x^3*(54 + 522*x + 1242*x^2 + 400*x^3 + 904*x^4 - 368*x^5 + 393*x^6 - 369*x^7 + 104*x^8 - 108*x^9 + 13*x^10 - 11*x^11) / (1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>14.
(End)

A187303 Number of 8-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.

Original entry on oeis.org

0, 0, 72, 2028, 12778, 37124, 81362, 145100, 231552, 338044, 465734, 613212, 780752, 968044, 1175118, 1401948, 1648534, 1914876, 2200974, 2506828, 2832438, 3177804, 3542926, 3927804, 4332438, 4756828, 5200974, 5664876, 6148534, 6651948, 7175118

Offset: 1

R. H. Hardin, Mar 08 2011

nonn

Row 8 of A187296.

			Some solutions for 4 X 4:
..0..1..0..0....1..6..5..0....0..6..5..7....6..0..7..0....8..7..6..2
..0..6..5..7....0..0..4..0....0..0..4..0....5..1..0..0....0..0..5..1
..3..2..4..0....2..7..3..8....2..1..3..8....4..0..8..0....0..0..4..3
..0..0..0..8....0..0..0..0....0..0..0..0....3..2..0..0....0..0..0..0

R. H. Hardin, Table of n, a(n) for n = 1..50

Cf. A187296.

Empirical: a(n) = 9878*n^2 - 79388*n + 143388 for n>13.
Conjectures from Colin Barker, Apr 23 2018: (Start)
G.f.: 2*x^3*(36 + 906*x + 3455*x^2 + 2401*x^3 + 3148*x^4 - 196*x^5 + 1607*x^6 - 1337*x^7 + 579*x^8 - 705*x^9 + 137*x^10 - 155*x^11 + 15*x^12 - 13*x^13) / (1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>16.
(End)

A187304 Number of 9-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.

Original entry on oeis.org

0, 0, 54, 2968, 25716, 87432, 209700, 399524, 668522, 1010572, 1430378, 1921532, 2485628, 3120300, 3825896, 4601996, 5448634, 6365780, 7353434, 8411596, 9540266, 10739444, 12009130, 13349324, 14760026, 16241236, 17792954, 19415180, 21107914

Offset: 1

R. H. Hardin Mar 08 2011

nonn

Row 9 of A187296

R. H. Hardin, Table of n, a(n) for n = 1..50

Empirical: a(n) = 35254*n^2 - 316744*n + 644876 for n>15

A187305 Number of 10-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.

Original entry on oeis.org

0, 0, 0, 3096, 44824, 187924, 505878, 1044900, 1847506, 2912204, 4254568, 5853664, 7715596, 9827696, 12192058, 14805028, 17667036, 20777536, 24136566, 27744092, 31600114, 35704632, 40057646, 44659156, 49509162, 54607664, 59954662, 65550156

Offset: 1

R. H. Hardin Mar 08 2011

nonn

Row 10 of A187296

R. H. Hardin, Table of n, a(n) for n = 1..50

Empirical: a(n) = 124248*n^2 - 1238146*n + 2807812 for n>17

cp's OEIS Frontend

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.

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A187298 Number of 3-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.

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A187299 Number of 4-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.

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A187300 Number of 5-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.

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A187301 Number of 6-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.

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A187302 Number of 7-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.

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A187303 Number of 8-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.

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A187304 Number of 9-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.

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Formula

A187305 Number of 10-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.

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