A203905 - OEIS

%I #6 Jul 12 2012 00:39:54

%S 1,0,0,1,1,1,0,0,0,0,1,1,2,1,1,0,0,0,0,0,0,1,1,2,2,2,1,1,0,0,0,0,0,0,

%T 0,0,1,1,2,2,3,2,2,1,1,0,0,0,0,0,0,0,0,0,0,1,1,2,2,3,3,3,2,2,1,1,0,0,

%U 0,0,0,0,0,0,0,0,0,0,1,1,2,2,3,3,4,3,3,2,2,1,1,0,0,0,0,0,0,0,0

%N Symmetric matrix based on (1,0,1,0,1,0,1,0,...), by antidiagonals.

%C Let s be the periodic sequence (1,0,1,0,1,0,...)  and let T be the infinite square matrix whose n-th row is formed by putting n-1 zeros before the terms of s.  Let T' be the transpose of T.  Then A203905 represents the matrix product M=T'*T.  M is the self-fusion matrix of s, as defined at A193722.  See A203906 for characteristic polynomials of principal submatrices of M, with interlacing zeros.

%e Northwest corner:

%e 1...0...1...0...1...0...1...0

%e 0...1...0...1...0...1...0...1

%e 1...0...2...0...2...0...2...0

%e 0...1...0...2...0...2...0...2

%e 1...0...2...0...3...0...3...0

%t t = {1, 0}; t1 = Flatten[{t, t, t, t, t, t, t, t, t, t}];

%t s[k_] := t1[[k]];

%t U = NestList[Most[Prepend[#, 0]] &, #,

%t Length[#] - 1] &[Table[s[k], {k, 1, 15}]];

%t L = Transpose[U]; M = L.U; TableForm[M] (* A203905 *)

%t m[i_, j_] := M[[i]][[j]];

%t Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]

%Y Cf. A203906, A202453.

%K tabl,sign

%O 1,13

%A _Clark Kimberling_, Jan 08 2012