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A203953 Symmetric matrix based on (1,2,1,2,1,2,...), by antidiagonals.

%I A203953 #6 Jul 12 2012 00:39:54
%S A203953 1,2,2,1,5,1,2,4,4,2,1,5,6,5,1,2,4,6,6,4,2,1,5,6,10,6,5,1,2,4,6,8,8,6,
%T A203953 4,2,1,5,6,10,11,10,6,5,1,2,4,6,8,10,10,8,6,4,2,1,5,6,10,11,15,11,10,
%U A203953 6,5,1,2,4,6,8,10,12,12,10,8,6,4,2,1,5,6,10,11,15,16,15,11,10
%N A203953 Symmetric matrix based on (1,2,1,2,1,2,...), by antidiagonals.
%C A203953 Let s be the periodic sequence (1,2,1,2,1,2,...) 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 A203951 represents the matrix product M=T'*T.  M is the self-fusion matrix of s, as defined at A193722.  See A203954 for characteristic polynomials of principal submatrices of M, with interlacing zeros.
%e A203953 Northwest corner:
%e A203953 1 2 1 2 1 2 1
%e A203953 2 5 4 5 4 5 4
%e A203953 1 3 6 6 6 6 6
%t A203953 t = {1, 2}; t1 = Flatten[{t, t, t, t, t, t, t, t, t, t}];
%t A203953 s[k_] := t1[[k]];
%t A203953 U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[
%t A203953    Table[s[k], {k, 1, 15}]];
%t A203953 L = Transpose[U]; M = L.U; TableForm[M]  (* A203953 *)
%t A203953 m[i_, j_] := M[[i]][[j]];
%t A203953 Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]
%Y A203953 Cf. A203954, A202453.
%K A203953 nonn,tabl
%O A203953 1,2
%A A203953 _Clark Kimberling_, Jan 08 2012