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 A204113 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the matrix at A204112, given by f(i,j) = gcd(F(i+1), F(j+1)), where F=A000045 (Fibonacci numbers). 3
 1, -1, 1, -3, 1, 2, -8, 6, -1, 8, -36, 35, -11, 1, 48, -232, 274, -116, 19, -1, 576, -2880, 3620, -1728, 358, -32, 1, 10368, -52992, 70632, -37192, 8906, -1016, 53, -1, 331776, -1716480, 2354112, -1294352, 332812, -42924, 2805 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS Let p(n)=p(n,x) be the characteristic polynomial of the n-th principal submatrix. The zeros of p(n) are real, and they interlace the zeros of p(n+1). See A202605 and A204016 for guides to related sequences. REFERENCES (For references regarding interlacing roots, see A202605.) LINKS Table of n, a(n) for n=1..42. EXAMPLE Top of the array: 1, -1; 1, -3, 1; 2, -8, 6, -1; 8, -36, 35, -11, 1; MATHEMATICA u[n_] := Fibonacci[n + 1] f[i_, j_] := GCD[u[i], u[j]]; m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}] TableForm[m[8]] (* 8 X 8 principal submatrix *) Flatten[Table[f[i, n + 1 - i], {n, 1, 15}, {i, 1, n}]] (* A204112 *) p[n_] := CharacteristicPolynomial[m[n], x]; c[n_] := CoefficientList[p[n], x] TableForm[Flatten[Table[p[n], {n, 1, 10}]]] Table[c[n], {n, 1, 12}] Flatten[%] (* A204113 *) TableForm[Table[c[n], {n, 1, 10}]] CROSSREFS Cf. A204112, A202605, A204016. Sequence in context: A060750 A204025 A204126 * A204128 A266272 A201677 Adjacent sequences: A204110 A204111 A204112 * A204114 A204115 A204116 KEYWORD tabl,sign AUTHOR Clark Kimberling, Jan 11 2012 STATUS approved

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Last modified February 29 21:35 EST 2024. Contains 370428 sequences. (Running on oeis4.)