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A255290 Number of 2's in expansion of F^n mod 3, where F = 1/(x*y)+2/y+x/y+2/x+2*x+y/x+2*y+x*y.

%I A255290 #5 Feb 21 2015 15:09:01
%S A255290 0,4,9,4,32,36,9,84,96,4,32,84,32,256,288,36,336,420,9,84,216,84,672,
%T A255290 744,96,852,1197,4,32,84,32,256,336,84,672,852,32,256,672,256,2048,
%U A255290 2304,288,2352,2544,36,336,864,336,2688
%N A255290 Number of 2's in expansion of F^n mod 3, where F = 1/(x*y)+2/y+x/y+2/x+2*x+y/x+2*y+x*y.
%C A255290 A255289 and A255290 together are a second mod 3 analog of A160239.
%e A255290 The pairs [no. of 1's, no. of 2's] are [1, 0], [4, 4], [12, 9], [4, 4], [32, 32], [48, 36], [12, 9], [84, 84], [117, 96], [4, 4], [32, 32], [84, 84], [32, 32], [256, 256], [300, 288], [48, 36], [336, 336], [324, 420], [12, 9], [84, 84], [225, 216], [84, 84], [672, 672], [792, 744], [117, 96], [852, 852], [876, 1197], ...
%p A255290 # C3 Counts 1's and 2's
%p A255290 C3 := proc(f) local c,ix,iy,f2,i,t1,t2,n1,n2;
%p A255290 f2:=expand(f) mod 3; n1:=0; n2:=0;
%p A255290 if whattype(f2) = `+` then
%p A255290 t1:=nops(f2);
%p A255290 for i from 1 to t1 do t2:=op(i, f2); ix:=degree(t2, x); iy:=degree(t2, y);
%p A255290 c:=coeff(coeff(t2,x,ix),y,iy);
%p A255290 if (c mod 3) = 1 then n1:=n1+1; else n2:=n2+1; fi; od: RETURN([n1,n2]);
%p A255290 else ix:=degree(f2, x); iy:=degree(f2, y);
%p A255290 c:=coeff(coeff(f2,x,ix),y,iy);
%p A255290 if (c mod 3) = 1 then n1:=n1+1; else n2:=n2+1; fi; RETURN([n1,n2]);
%p A255290 fi;
%p A255290 end;
%p A255290 F2:=1/(x*y)+2/y+x/y+2/x+2*x+y/x+2*y+x*y mod 3;
%p A255290 g:=(F,n)->expand(F^n) mod 3;
%p A255290 [seq(C3(g(F2,n))[2],n=0..60)];
%Y A255290 Cf. A160239, A255287-A255294.
%K A255290 nonn
%O A255290 0,2
%A A255290 _N. J. A. Sloane_, Feb 21 2015