This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A343530 #37 Jun 17 2021 13:58:53 %S A343530 0,1,12,10,13,16,35,51,56,90,42,84,99,129,156,30,220,184,201,79,321, %T A343530 25,424,301,389,29,32,311,328,186,129,42,101,97,144,52,534,83,506,885, %U A343530 233,472,43,410,145,210,482,51,57,144,53,60,148,248,83,80,180,72,55 %N A343530 Number of steps before being trapped for a knight moving on a square-spiral base-n numbered board when stepping to the closest unvisited square which contains a number that shares no digit with the number of the current square. If two or more such squares are the same distance away the one with the smaller number is chosen. %H A343530 Rémy Sigrist, <a href="/A343530/b343530.txt">Table of n, a(n) for n = 2..3500</a> %H A343530 Scott R. Shannon, <a href="/A343530/a343530.png">Image of the path for a(4) = 12</a>. In this and other images the starting square is highlighted in white, the visited squares, numbered in base n, in yellow, the final square in red, while the path is colored across the spectrum to show the relative step ordering. %H A343530 Scott R. Shannon, <a href="/A343530/a343530_1.png">Image of the path for a(8) = 35</a>. %H A343530 Scott R. Shannon, <a href="/A343530/a343530_2.png">Image of the path for a(10) = 56</a> %H A343530 Scott R. Shannon, <a href="/A343530/a343530_3.png">Image of the path for a(16) = 156</a>. %H A343530 Scott R. Shannon, <a href="/A343530/a343530_4.png">Image of the path for a(24) = 424</a>. %H A343530 Scott R. Shannon, <a href="/A343530/a343530_5.png">Image of the path for a(36) = 144</a>. %H A343530 Rémy Sigrist, <a href="/A343530/a343530.gp.txt">PARI program for A343530</a> %F A343530 a(n) = 2015 for any n >= 2979. - _Rémy Sigrist_, Jun 16 2021 %e A343530 The board in base 10 is numbered with the square spiral: %e A343530 . %e A343530 17--16--15--14--13 . %e A343530 | | . %e A343530 18 5---4---3 12 29 %e A343530 | | | | | %e A343530 19 6 1---2 11 28 %e A343530 | | | | %e A343530 20 7---8---9--10 27 %e A343530 | | %e A343530 21--22--23--24--25--26 %e A343530 . %e A343530 a(2) = 0 as on a base-2 numbered spiral all surrounding squares contain a 1 digit in their number thus, as the knight starts on the square numbered 1, it has no square to move to which does not contain a 1 digit. %e A343530 a(3) = 1 as on a base-3 numbered board there are two squares the knight can step to which do not contain a 1 digit -- the squares numbered 200_3 = 18 and 220_3 = 24. The knight steps to 200_3 as it is the lowest numbered square, but after that there are no surrounding unvisited squares the knight can step to which do not contain the digit 0 or 2. %e A343530 a(4) = 12 as on a base-4 numbered board the knight steps to squares 22_4 = 10, 3_4 = 3, 12_4 = 6, 33_4 = 15, 2_4 = 2, 11_4 = 5, 20_4 = 8, 111_4 = 21, 220_4 = 40, 13_4 = 7, 222_4 = 42, 103_4 = 19. The knight is then trapped as no unvisited square containing only the digit 2 is one knight step away. %e A343530 See the linked images for other examples. %o A343530 (PARI) See Links section. %Y A343530 Cf. A344325, A343563, A326918, A328894, A326916, A316667. %K A343530 nonn,base %O A343530 2,3 %A A343530 _Scott R. Shannon_ and _Eric Angelini_, Apr 19 2021 %E A343530 More terms from _Rémy Sigrist_, Jun 16 2021