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A255288 Number of 2's in expansion of F^n mod 3, where F = 1/(x*y)+1/y+x/y+1/x+x+y/x+y+x*y.

%I A255288 #9 Feb 21 2015 15:01:52
%S A255288 0,0,13,0,0,32,13,104,112,0,0,104,0,0,184,32,256,296,13,104,208,104,
%T A255288 832,836,112,896,1081,0,0,104,0,0,256,104,832,896,0,0,832,0,0,1400,
%U A255288 184,1472,1768,32,256,932,256,2048,2692,296
%N A255288 Number of 2's in expansion of F^n mod 3, where F = 1/(x*y)+1/y+x/y+1/x+x+y/x+y+x*y.
%C A255288 A255287 and A255288 together are a mod 3 analog of A160239.
%e A255288 The pairs [no. of 1's, no. of 2's] are [1, 0], [8, 0], [8, 13], [8, 0], [64, 0], [52, 32], [8, 13], [64, 104], [101, 112], [8, 0], [64, 0], [64, 104], [64, 0], [512, 0], [404, 184], [52, 32], [416, 256], [448, 296], [8, 13], [64, 104], [233, 208], [64, 104], [512, 832], [700, 836], [101, 112], [808, 896], [992, 1081], ...
%p A255288 # C3 Counts 1's and 2's
%p A255288 C3 := proc(f) local c,ix,iy,f2,i,t1,t2,n1,n2;
%p A255288 f2:=expand(f) mod 3; n1:=0; n2:=0;
%p A255288 if whattype(f2) = `+` then
%p A255288 t1:=nops(f2);
%p A255288 for i from 1 to t1 do t2:=op(i, f2); ix:=degree(t2, x); iy:=degree(t2, y);
%p A255288 c:=coeff(coeff(t2,x,ix),y,iy);
%p A255288 if (c mod 3) = 1 then n1:=n1+1; else n2:=n2+1; fi; od: RETURN([n1,n2]);
%p A255288 else ix:=degree(f2, x); iy:=degree(f2, y);
%p A255288 c:=coeff(coeff(f2,x,ix),y,iy);
%p A255288 if (c mod 3) = 1 then n1:=n1+1; else n2:=n2+1; fi; RETURN([n1,n2]);
%p A255288 fi;
%p A255288 end;
%p A255288 F1:=1/(x*y)+1/y+x/y+1/x+x+y/x+y+x*y mod 3;
%p A255288 g:=(F,n)->expand(F^n) mod 3;
%p A255288 [seq(C3(g(F1,n))[2],n=0..60)];
%Y A255288 Cf. A160239, A255287-A255294.
%K A255288 nonn
%O A255288 0,3
%A A255288 _N. J. A. Sloane_, Feb 21 2015