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A180319 Total number of possible standard knight moves on an n X 2n chessboard, if the knight is placed anywhere.

%I A180319 #12 Aug 21 2025 06:43:58
%S A180319 0,8,52,128,236,376,548,752,988,1256,1556,1888,2252,2648,3076,3536,
%T A180319 4028,4552,5108,5696,6316,6968,7652,8368,9116,9896,10708,11552,12428,
%U A180319 13336,14276,15248,16252,17288,18356,19456,20588,21752,22948,24176,25436
%N A180319 Total number of possible standard knight moves on an n X 2n chessboard, if the knight is placed anywhere.
%C A180319 a(n) counts every possible moves of a standard chess knight placed anywhere.
%F A180319 Conjecture: a(n) = 4*(4-9*n+4*n^2) for n>1. a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>4. G.f.: 4*x^2*(2+7*x-x^2)/(1-x)^3. - _Colin Barker_, Mar 11 2012
%e A180319 For example, in usual chessboard 8X8 a knight in a corner has only 2 moves, in a central square it has the maximum number of moves: 8.
%e A180319 Summing all over the 64 squares we have 336 possible moves.
%e A180319 Instead on a chessboard 4x8 the number is reduced:
%e A180319   -----------------
%e A180319   |2|3|4|4|4|4|3|2|
%e A180319   -----------------
%e A180319   |3|4|6|6|6|6|4|3|
%e A180319   -----------------
%e A180319   |3|4|6|6|6|6|4|3|
%e A180319   -----------------
%e A180319   |2|3|4|4|4|4|3|2|
%e A180319   -----------------
%e A180319 the total number is 128.
%Y A180319 Cf. A035008.
%K A180319 easy,nonn
%O A180319 1,2
%A A180319 Graziano Aglietti (mg5055(AT)mclink.it), Aug 27 2010