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%I #7 Jul 12 2012 00:39:54
%S 1,0,0,0,1,0,1,0,0,1,0,0,1,0,0,0,1,0,0,1,0,1,0,0,2,0,0,1,0,0,1,0,0,1,
%T 0,0,0,1,0,0,2,0,0,1,0,1,0,0,2,0,0,2,0,0,1,0,0,1,0,0,2,0,0,1,0,0,0,1,
%U 0,0,2,0,0,2,0,0,1,0,1,0,0,2,0,0,3,0,0,2,0,0,1,0,0,1,0,0,2,0,0
%N Symmetric matrix based on (1,0,0,1,0,0,1,0,0,...), by antidiagonals.
%C Let s be the periodic sequence (1,0,0,1,0,0,...) 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 A203945 represents the matrix product M=T'*T. M is the self-fusion matrix of s, as defined at A193722. See A203946 for characteristic polynomials of principal submatrices of M, with interlacing zeros.
%e Northwest corner:
%e 1...0...0...1...0...0...1
%e 0...1...0...0...1...0...0
%e 0...0...1...0...0...1...0
%e 1...0...0...2...0...0...2
%e 0...1...0...0...2...0...0
%t t = {1, 0, 0}; t1 = Flatten[{t, t, t, t, t, t, t, t}];
%t s[k_] := t1[[k]];
%t U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[
%t Table[s[k], {k, 1, 15}]];
%t L = Transpose[U]; M = L.U; TableForm[M]
%t m[i_, j_] := M[[i]][[j]];
%t Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]
%Y Cf. A203946, A202453.
%K nonn,tabl
%O 1,25
%A _Clark Kimberling_, Jan 08 2012
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