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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. 1
 2084, 124561, 1756923, 21375782, 48176535, 128322490, 196727321, 230310289, 606217402, 2856313870, 244655558, 659075420, 586292888, 1646774611, 1018215514, 719687377, 564513339, 2779028614, 298995630, 1641747842, 414061107, 1467655587, 584309414, 1584716050 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 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. LINKS Table of n, a(n) for n=1..24. 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) EXAMPLE 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 CROSSREFS Cf. A316884, A316967, A316667, A316328, A317106, A317105, A317416, A317415, A317438, A317437. Cf. A323469, A323470, A323471, A323472. Sequence in context: A323750 A323471 A251225 * A201917 A250076 A255146 Adjacent sequences: A306418 A306419 A306420 * A306422 A306423 A306424 KEYWORD nonn AUTHOR Scott R. Shannon, Feb 14 2019 STATUS approved

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Last modified June 10 11:18 EDT 2023. Contains 363205 sequences. (Running on oeis4.)