A306421 End squares for a trapped knight moving on a spirally numbered 2D grid where each square can be visited n times.

2084, 124561, 1756923, 21375782, 48176535, 128322490, 196727321, 230310289, 606217402, 2856313870, 244655558, 659075420, 586292888, 1646774611, 1018215514, 719687377, 564513339, 2779028614, 298995630, 1641747842, 414061107, 1467655587, 584309414, 1584716050

Offset: 1

Author

Scott R. Shannon, Feb 14 2019

Keywords

• nonn

Comments

For a knight (a (1,2) leaper) starting at square 1 and moving on a spirally numbered 2D grid to the lowest-numbered available square at each step (see A316667), a(n) is the number of the square at which the knight is trapped if it is allowed to visit each square no more than n times -- the knight is not trapped until each of the 8 surrounding squares to which it can leap has been visited n times. The choice of the square to which it goes at each step is determined solely by the square with the lowest spiral number, as long as it has been visited fewer than n times. This is an infinite sequence, although end squares beyond a(35) are currently unknown.

Examples

			For n = 1, the knight becomes trapped at square 2084 (see A316667). The following table gives the corresponding values for n = 1 through 35:
.
     | Square at which | Number of steps
     |  the knight is  | before the
   n |     trapped     | knight is trapped
  ---+-----------------+--------------
   1 |         2084    |          2016 (A316667)
   2 |       124561    |        244273
   3 |      1756923    |       4737265
   4 |     21375782    |      98374180
   5 |     48176535    |     258063291
   6 |    128322490    |     836943142
   7 |    196727321    |    1531051657
   8 |    230310289    |    1897092533
   9 |    606217402    |    5253106114
  10 |   2856313870    |   27296872250
  11 |    244655558    |    2772304666
  12 |    659075420    |    8437814958
  13 |    586292888    |    7875951360
  14 |   1646774611    |   24511621133
  15 |   1018215514    |   15493169264
  16 |    719687377    |   11643899003
  17 |    564513339    |    9593491769
  18 |   2779028614    |   49835086546
  19 |    298995630    |    5734502340
  20 |   1641747842    |   33370972720
  21 |    414061107    |    8844741817
  22 |   1467655587    |   32843399937
  23 |    584309414    |   13583967470
  24 |   1584716050    |   37945957450
  25 |   2544445470    |   62083869640
  26 |   4796115990    |  125967045044
  27 |   1881606731    |   51291895045
  28 |   1321212795    |   37635024035
  29 |   6693611092    |  196994700434
  30 |    687619472    |   19985943874
  31 |   1495794139    |   45392651369
  32 |   6677258413    |  213836002227
  33 |   6451059544    |  219770103702
  34 |   7958333435    |  277128625469
  35 |  13924943879    |  485324576539
		

Links

• Scott R. Shannon, Simplified Java code for producing the series
• Scott R. Shannon, Visited positions for n=3. For clarity only the visited positions are shown. Blue=3 visits, Green=2 visits, White=1 visit. Red is the final square (near top right corner). Note that the internal positions are all visited the maximum 3 times, and that the overall shape becomes an 'indented square' -- this pattern becomes more pronounced as n increases. Likewise the maximum visited x and y distances relative to the central square approach equality as n increases e.g. for n=35 both the maximum x and y visited distances are 59855.
• N. J. A. Sloane and Brady Haran, The Trapped Knight, Numberphile video (2019)

Crossrefs

Cf. A316884, A316967, A316667, A316328, A317106, A317105, A317416, A317415, A317438, A317437.

Cf. A323469, A323470, A323471, A323472.