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A377018 a(n) is the number of paths of a knight on square a1 to reach a position outside an 8 X 8 chessboard after n steps.

%I A377018 #26 May 25 2025 23:25:35
%S A377018 6,4,32,108,880,4420,29560,167068,1041440,6128772,37298680,222571260,
%T A377018 1343492128,8055277508,48487848472,291196932444,1751154444320,
%U A377018 10522542700868,63258161555448,380185630909692,2285299322957920,13735692739737604,82562224208247576,496247752160871132
%N A377018 a(n) is the number of paths of a knight on square a1 to reach a position outside an 8 X 8 chessboard after n steps.
%C A377018 a(n)/8^n is the probability that the knight falls off the chess board at step n. The average number of steps it takes the knight to fall off the board is Sum_{n>0} n*a(n)/8^n = A376736(8)/A376737(8) = 101338/51901 or approximately 1.953 steps.
%C A377018 Because of the mirror symmetry of the problem to the board diagonal, all terms are even.
%H A377018 Ruediger Jehn, <a href="/A377018/b377018.txt">Table of n, a(n) for n = 1..30</a>
%H A377018 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (3,27,-29,-162,42,276,16,-96).
%F A377018 G.f.: 2*x*(3 - 7*x - 71*x^2 + 39*x^3 + 390*x^4 + 94*x^5 - 484*x^6 - 240*x^7)/(1 - 3*x - 27*x^2 + 29*x^3 + 162*x^4 - 42*x^5 - 276*x^6 - 16*x^7 + 96*x^8). - _Andrew Howroyd_, Oct 16 2024
%F A377018 a(n) ~ 0.10036158347592796... * 6.01066058303935...^n. - _Ruediger Jehn_, Nov 06 2024
%e A377018 a(2) = 4. Starting on square a1 there are 4 paths for a knight to leave the chess board in two steps: b3-z2, b3-z4, c2-b0 and c2-d0 (z being the file left of the a-file).
%t A377018 LinearRecurrence[{3, 27, -29, -162, 42, 276, 16, -96}, {6, 4, 32, 108, 880, 4420, 29560, 167068}, 24] (* _Hugo Pfoertner_, Oct 16 2024 *)
%o A377018 (PARI) Vec(2*(3 - 7*x - 71*x^2 + 39*x^3 + 390*x^4 + 94*x^5 - 484*x^6 - 240*x^7)/(1 - 3*x - 27*x^2 + 29*x^3 + 162*x^4 - 42*x^5 - 276*x^6 - 16*x^7 + 96*x^8) + O(x^30)) \\ _Andrew Howroyd_, Oct 16 2024
%Y A377018 Cf. A376736, A376737, A376837, A378902 (for a king).
%K A377018 nonn,easy
%O A377018 1,1
%A A377018 _Ruediger Jehn_, Oct 13 2024