A120854 Matrix log of A117939, read by rows, consisting only of 0's, 3's and signed 2's.
0, 2, 0, 3, -2, 0, 2, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 3, -2, 0, 3, 0, 0, -2, 0, 0, 0, 0, 3, 0, 0, -2, 0, 2, 0, 0, 0, 3, 0, 0, -2, 3, -2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 3, -2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 3, -2, 0, 0, 0, 0, 0
Offset: 0
Examples
Triangle begins: 0; 2, 0; 3,-2, 0; 2, 0, 0, 0; 0, 2, 0, 2, 0; 0, 0, 2, 3,-2, 0; 3, 0, 0,-2, 0, 0, 0; 0, 3, 0, 0,-2, 0, 2, 0; 0, 0, 3, 0, 0,-2, 3,-2, 0; 2, 0, 0, 0, 0, 0, 0, 0, 0, 0; ... Matrix exponentiation gives A117939: 1; 2, 1; 1,-2, 1; 2, 0, 0, 1; 4, 2, 0, 2, 1; 2,-4, 2, 1,-2, 1; 1, 0, 0,-2, 0, 0, 1; 2, 1, 0,-4,-2, 0, 2, 1; 1,-2, 1,-2, 4,-2, 1,-2, 1; ... and A117939 is the matrix square of A117947: 1; 1, 1; 1,-1, 1; 1, 0, 0, 1; 1, 1, 0, 1, 1; 1,-1, 1, 1,-1, 1; 1, 0, 0,-1, 0, 0, 1; 1, 1, 0,-1,-1, 0, 1, 1; 1,-1, 1,-1, 1,-1, 1,-1, 1; ...
Programs
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PARI
/* Generated as the Matrix LOG of A117939: */ T(n,k)=local(M=matrix(n+1,n+1,r,c,(binomial(r-1,c-1)+1)%3-1)^2, L=sum(i=1,#M,-(M^0-M)^i/i));return(L[n+1,k+1])
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PARI
/* Generated as the Ternary Fractal: */ T(n,k)=local(r=n,c=k,s=floor(log(n+1)/log(3))+1,u=vector(s),v=vector(s),d,e); if(n<=k,0,if(n<3&k<3,[0,0,0;2,0,0;3,-2,0][n+1,k+1], for(i=1,#u,u[i]=r%3;r=r\3);for(i=1,#v,v[i]=c%3;c=c\3); d=0;for(i=1,#v,if(u[i]!=v[i],d+=1;e=i));if(d==1,T(u[e],v[e]),0)))
Formula
Ternary fractal, T(3*n,3*k) = T(n,k), defined by: T(n,k) = 0 if n<=k or when more than 1 digit differs between the ternary expansions of n and k; else T(n,k) = T(m,j) where the only ternary digits of n, k, that differ are m, j, respectively and T(1,0)=2, T(2,1)=-2, T(2,0)=3.
Comments