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

%I A306421 #71 Sep 26 2019 11:03:13
%S A306421 2084,124561,1756923,21375782,48176535,128322490,196727321,230310289,
%T A306421 606217402,2856313870,244655558,659075420,586292888,1646774611,
%U A306421 1018215514,719687377,564513339,2779028614,298995630,1641747842,414061107,1467655587,584309414,1584716050
%N A306421 End squares for a trapped knight moving on a spirally numbered 2D grid where each square can be visited n times.
%C A306421 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.
%H A306421 Scott R. Shannon, <a href="/A306421/a306421_1.java.txt">Simplified Java code for producing the series</a>
%H A306421 Scott R. Shannon, <a href="/A306421/a306421.png">Visited positions for n=3</a>. 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.
%H A306421 N. J. A. Sloane and Brady Haran, <a href="https://www.youtube.com/watch?v=RGQe8waGJ4w">The Trapped Knight</a>, Numberphile video (2019)
%e A306421 For n = 1, the knight becomes trapped at square 2084 (see A316667). The following table gives the corresponding values for n = 1 through 35:
%e A306421 .
%e A306421      | Square at which | Number of steps
%e A306421      |  the knight is  | before the
%e A306421    n |     trapped     | knight is trapped
%e A306421   ---+-----------------+--------------
%e A306421    1 |         2084    |          2016 (A316667)
%e A306421    2 |       124561    |        244273
%e A306421    3 |      1756923    |       4737265
%e A306421    4 |     21375782    |      98374180
%e A306421    5 |     48176535    |     258063291
%e A306421    6 |    128322490    |     836943142
%e A306421    7 |    196727321    |    1531051657
%e A306421    8 |    230310289    |    1897092533
%e A306421    9 |    606217402    |    5253106114
%e A306421   10 |   2856313870    |   27296872250
%e A306421   11 |    244655558    |    2772304666
%e A306421   12 |    659075420    |    8437814958
%e A306421   13 |    586292888    |    7875951360
%e A306421   14 |   1646774611    |   24511621133
%e A306421   15 |   1018215514    |   15493169264
%e A306421   16 |    719687377    |   11643899003
%e A306421   17 |    564513339    |    9593491769
%e A306421   18 |   2779028614    |   49835086546
%e A306421   19 |    298995630    |    5734502340
%e A306421   20 |   1641747842    |   33370972720
%e A306421   21 |    414061107    |    8844741817
%e A306421   22 |   1467655587    |   32843399937
%e A306421   23 |    584309414    |   13583967470
%e A306421   24 |   1584716050    |   37945957450
%e A306421   25 |   2544445470    |   62083869640
%e A306421   26 |   4796115990    |  125967045044
%e A306421   27 |   1881606731    |   51291895045
%e A306421   28 |   1321212795    |   37635024035
%e A306421   29 |   6693611092    |  196994700434
%e A306421   30 |    687619472    |   19985943874
%e A306421   31 |   1495794139    |   45392651369
%e A306421   32 |   6677258413    |  213836002227
%e A306421   33 |   6451059544    |  219770103702
%e A306421   34 |   7958333435    |  277128625469
%e A306421   35 |  13924943879    |  485324576539
%Y A306421 Cf. A316884, A316967, A316667, A316328, A317106, A317105, A317416, A317415, A317438, A317437.
%Y A306421 Cf. A323469, A323470, A323471, A323472.
%K A306421 nonn
%O A306421 1,1
%A A306421 _Scott R. Shannon_, Feb 14 2019