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A051910 Triangle T(n,m) = Nim-product of n and m, read by rows, 0<=m<=n.

%I A051910 #13 Mar 30 2012 16:48:43
%S A051910 0,0,1,0,2,3,0,3,1,2,0,4,8,12,6,0,5,10,15,2,7,0,6,11,13,14,8,5,0,7,9,
%T A051910 14,10,13,3,4,0,8,12,4,11,3,7,15,13,0,9,14,7,15,6,1,8,5,12,0,10,15,5,
%U A051910 3,9,12,6,1,11,14,0,11,13,6,7,12,10,1,9,2,4,15
%N A051910 Triangle T(n,m) = Nim-product of n and m, read by rows, 0<=m<=n.
%D A051910 E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 60.
%D A051910 J. H. Conway, On Numbers and Games, Academic Press, p. 52.
%H A051910 R. J. Mathar, <a href="/A051910/b051910.txt">Table of n, a(n) for n = 0..3320</a>
%H A051910 <a href="/index/Ni#Nimmult">Index entries for sequences related to Nim-multiplication</a>
%F A051910 T(n,m) = A051775(n,m) = A051776(n,m).
%e A051910 Triangle starts
%e A051910 0;
%e A051910 0, 1;
%e A051910 0, 2,  3;
%e A051910 0, 3,  1,  2;
%e A051910 0, 4,  8, 12,  6;
%e A051910 0, 5, 10, 15,  2,  7;
%e A051910 0, 6, 11, 13, 14,  8, 5;
%e A051910 0, 7,  9, 14, 10, 13, 3,  4;
%e A051910 0, 8, 12,  4, 11,  3, 7, 15, 13;
%p A051910 We continue from A003987: to compute a Nim-multiplication table using (a) an addition table AT := array(0..NA, 0..NA) and (b) a nimsum procedure for larger values; MT := array(0..N,0..N); for a from 0 to N do MT[a,0] := 0; MT[0,a] := 0; MT[a,1] := a; MT[1,a] := a; od: for a from 2 to N do for b from a to N do t1 := {}; for i from 0 to a-1 do for j from 0 to b-1 do u1 := MT[i,b]; u2 := MT[a,j];
%p A051910 if u1<=NA and u2<=NA then u12 := AT[u1,u2]; else u12 := nimsum(u1,u2); fi; u3 := MT[i,j]; if u12<=NA and u3<=NA then u4 := AT[u12,u3]; else u4 := nimsum(u12,u3); fi; t1 := { op(t1), u4}; #t1 := { op(t1), AT[ AT[ MT[i,b], MT[a,j] ], MT[i,j] ] }; od; od;
%p A051910 t2 := sort(convert(t1,list)); j := nops(t2); for i from 1 to nops(t2) do if t2[i] <> i-1 then j := i-1; break; fi; od; MT[a,b] := j; MT[b,a] := j; od; od;
%Y A051910 Cf. A051776, A003987, A051775, A051911.
%K A051910 tabl,nonn,easy,nice
%O A051910 0,5
%A A051910 _N. J. A. Sloane_, Dec 20 1999