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A204030 Symmetric matrix based on f(i,j) = gcd(i+1, j+1), by antidiagonals. 3
2, 1, 1, 2, 3, 2, 1, 1, 1, 1, 2, 1, 4, 1, 2, 1, 3, 1, 1, 3, 1, 2, 1, 2, 5, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 1, 6, 1, 4, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 7, 2, 1, 2, 1, 2, 1, 3, 1, 5, 3, 1, 1, 3, 5, 1, 3, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

A204030 represents the matrix M given by f(i,j) = gcd(i+1, j+1) for i >= 1 and j >= 1.  See A204031 for characteristic polynomials of principal submatrices of M, with interlacing zeros.  See A204016 for a guide to other choices of M.

LINKS

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

EXAMPLE

Northwest corner:

  2 1 2 1 2 1 2 1

  1 3 1 1 3 1 1 3

  2 1 4 1 2 1 4 1

  1 1 1 5 1 1 1 1

  2 3 2 1 6 1 2 3

  1 1 1 1 1 7 1 1

MATHEMATICA

f[i_, j_] := GCD[i + 1, j + 1];

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}]]  (* A204030 *)

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[%]                 (* A204111 *)

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

CROSSREFS

Cf. A204111, A204016, A202453.

Sequence in context: A072782 A337014 A122563 * A234503 A333267 A236325

Adjacent sequences:  A204027 A204028 A204029 * A204031 A204032 A204033

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Jan 11 2012

STATUS

approved

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Last modified March 8 01:41 EST 2021. Contains 341934 sequences. (Running on oeis4.)