A204116 - OEIS

%I #9 Aug 02 2019 04:12:34

%S 1,1,1,1,3,1,1,1,1,1,1,3,7,3,1,1,1,1,1,1,1,1,3,1,15,1,3,1,1,1,7,1,1,7,

%T 1,1,1,3,1,3,31,3,1,3,1,1,1,1,1,1,1,1,1,1,1,1,3,7,15,1,63,1,15,7,3,1,

%U 1,1,1,1,1,1,1,1,1,1,1,1,1,3,1,3,1,3,127,3,1,3,1,3,1,1,1,7,1

%N Symmetric matrix based on f(i,j) = gcd(2^i-1, 2^j-1), by antidiagonals.

%C A204116 represents the matrix M given by f(i,j) = gcd(2^i-1, 2^j-1) for i >= 1 and j >= 1.  See A204117 for characteristic polynomials of principal submatrices of M, with interlacing zeros.  See A204016 for a guide to other choices of M.

%e Northwest corner:

%e   1  1  1  1

%e   1  3  1  3

%e   1  1  7  1

%e   1  3  1 15

%t f[i_, j_] := GCD[2^i - 1, 2^j - 1];

%t m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]

%t TableForm[m[8]] (* 8 X 8 principal submatrix *)

%t Flatten[Table[f[i, n + 1 - i],

%t   {n, 1, 15}, {i, 1, n}]]  (* A204116 *)

%t p[n_] := CharacteristicPolynomial[m[n], x];

%t c[n_] := CoefficientList[p[n], x]

%t TableForm[Flatten[Table[p[n], {n, 1, 10}]]]

%t Table[c[n], {n, 1, 12}]

%t Flatten[%]                 (* A204117 *)

%t TableForm[Table[c[n], {n, 1, 10}]]

%Y Cf. A204117, A204016, A202453.

%K nonn,tabl

%O 1,5

%A _Clark Kimberling_, Jan 11 2012