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A204112 Symmetric matrix based on f(i,j) = gcd(F(i+1), F(j+1)), where F=A000045 (Fibonacci numbers), by antidiagonals.

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%I A204112 #9 Aug 02 2019 04:12:28
%S A204112 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 A204112 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 A204112 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 A204112 Symmetric matrix based on f(i,j) = gcd(F(i+1), F(j+1)), where F=A000045 (Fibonacci numbers), by antidiagonals.
%C A204112 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 A204112 Northwest corner:
%e A204112   1  1  1  1  1  1
%e A204112   1  2  1  1  2  1
%e A204112   1  1  3  1  1  1
%e A204112   1  1  1  5  1  1
%e A204112   1  2  1  1  8  1
%e A204112   1  1  1  1  1 13
%t A204112 u[n_] := Fibonacci[n + 1]
%t A204112 f[i_, j_] := GCD[u[i], u[j]];
%t A204112 m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t A204112 TableForm[m[8]] (* 8 X 8 principal submatrix *)
%t A204112 Flatten[Table[f[i, n + 1 - i],
%t A204112   {n, 1, 15}, {i, 1, n}]]    (* A204112 *)
%t A204112 p[n_] := CharacteristicPolynomial[m[n], x];
%t A204112 c[n_] := CoefficientList[p[n], x]
%t A204112 TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
%t A204112 Table[c[n], {n, 1, 12}]
%t A204112 Flatten[%]                   (* A204113 *)
%t A204112 TableForm[Table[c[n], {n, 1, 10}]]
%Y A204112 Cf. A204113, A204016, A202453.
%K A204112 nonn,tabl
%O A204112 1,5
%A A204112 _Clark Kimberling_, Jan 11 2012