%I #9 Aug 02 2019 04:12:28
%S 1,1,1,1,2,1,1,1,1,1,1,1,3,1,1,1,2,1,1,2,1,1,1,1,5,1,1,1,1,1,1,1,1,1,
%T 1,1,1,2,3,1,8,1,3,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,13,1,1,1,1,1,1,2,
%U 1,5,2,1,1,2,5,1,2,1,1,1,3,1,1,1,21,1,1,1,3,1,1,1,1,1,1,1,1,1
%N Symmetric matrix based on f(i,j) = gcd(F(i+1), F(j+1)), where F=A000045 (Fibonacci numbers), by antidiagonals.
%C A204112 represents the matrix M given by f(i,j) = gcd(F(i+1), F(j+1)) for i >= 1 and j >= 1. See A204113 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 1 1
%e 1 2 1 1 2 1
%e 1 1 3 1 1 1
%e 1 1 1 5 1 1
%e 1 2 1 1 8 1
%e 1 1 1 1 1 13
%t u[n_] := Fibonacci[n + 1]
%t f[i_, j_] := GCD[u[i], u[j]];
%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}]] (* A204112 *)
%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[%] (* A204113 *)
%t TableForm[Table[c[n], {n, 1, 10}]]
%Y Cf. A204113, A204016, A202453.
%K nonn,tabl
%O 1,5
%A _Clark Kimberling_, Jan 11 2012