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%I #11 Apr 26 2026 17:55:57
%S 1,-1,2,-2,-3,3,-4,0,0,4,-5,5,-6,0,0,6,0,0,0,0,7,8,0,0,-7,-8,0,0,0,0,
%T 9,10,0,0,-9,-10,11,-11,0,0,12,-12,13,0,-13,0,0,14,15,-14,16,0,0,0,0,
%U -15,17,-16,-17,0,0,18,0,-18,-19,19,-20,20,-21,21,22
%N Consider the square spiral with its cells numbered starting at 0. Two players, Black and Red, take turns. When it is Black's turn, he places a knight at the smallest unoccupied cell not attacked by an existing Red knight, and when it is Red's turn, she places a knight at the smallest unoccupied cell not attacked by an existing Black knight. a(n) = +k if the n-th cell is occupied by the k-th Black knight, a(n) = -k if the n-th cell is occupied by the k-th Red knight, a(n) = 0 otherwise.
%C Each nonzero integer appears exactly once.
%H Rémy Sigrist, <a href="/A395486/b395486.txt">Table of n, a(n) for n = 0..10000</a>
%F a(A392177(k)) = +k.
%F a(A392178(k)) = -k.
%F a(A392179(k)) = 0.
%F A392180(n) = Sum_{k = 0..n} sign(a(k)).
%e The first terms alongside the square spiral are:
%e 0__32___0___0_-30_-29___0__31___0___0_-28
%e | |
%e 0 -19_-18___0__18___0___0_-17_-16__17 30
%e | | | |
%e 0 19 11_-10__-9___0___0__10___9 -15 -27
%e | | | | | |
%e 0 -20 -11 0___6___0___0__-6 0 0 29
%e | | | | | | | |
%e 0 20 0 0 -3__-2___2 5 0 0 -26
%e | | | | | | | | | |
%e 0 -21 0 0 3 1__-1 -5 0 0 28
%e | | | | | | | | |
%e 0 21 12 0 -4___0___0___4 0 0 -25
%e | | | | | | |
%e 0 22 -12 7___8___0___0__-7__-8 16 27
%e | | | | |
%e 0 23 13___0_-13___0___0__14__15_-14 -24
%e | | |
%e 33 -22___0___0__24__25___0_-23___0___0__26
%e |
%e 34__35__36___0_-31_-32__37__38___0___0_-33
%o (C#) // Use program provided in A392177 with the 3 arguments "- - layout".
%Y Cf. A392177, A392178, A392179, A392180.
%K sign
%O 0,3
%A _Rémy Sigrist_, Apr 25 2026