A203905 Symmetric matrix based on (1,0,1,0,1,0,1,0,...), by antidiagonals.

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, 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, 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

Offset

1,13

Comments

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.

Links

Table of n, a(n) for n=1..99.

Example

Northwest corner:
1...0...1...0...1...0...1...0
0...1...0...1...0...1...0...1
1...0...2...0...2...0...2...0
0...1...0...2...0...2...0...2
1...0...2...0...3...0...3...0

Mathematica

t = {1, 0}; t1 = Flatten[{t, t, t, t, t, t, t, t, t, t}];
s[k_] := t1[[k]];
U = NestList[Most[Prepend[#, 0]] &, #,
Length[#] - 1] &[Table[s[k], {k, 1, 15}]];
L = Transpose[U]; M = L.U; TableForm[M] (* A203905 *)
m[i_, j_] := M[[i]][[j]];
Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]

Crossrefs

Cf. A203906, A202453.

Sequence in context: A016408 A316868 A106822 * A309142 A064532 A321922

Adjacent sequences: A203902 A203903 A203904 * A203906 A203907 A203908

Keyword

tabl,sign

Author

Clark Kimberling, Jan 08 2012

Status

approved