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A204154 Symmetric matrix based on f(i,j) = max(2i-j, 2j-i), by antidiagonals. 7
1, 3, 3, 5, 2, 5, 7, 4, 4, 7, 9, 6, 3, 6, 9, 11, 8, 5, 5, 8, 11, 13, 10, 7, 4, 7, 10, 13, 15, 12, 9, 6, 6, 9, 12, 15, 17, 14, 11, 8, 5, 8, 11, 14, 17, 19, 16, 13, 10, 7, 7, 10, 13, 16, 19, 21, 18, 15, 12, 9, 6, 9, 12, 15, 18, 21, 23, 20, 17, 14, 11, 8, 8, 11, 14, 17, 20 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

A204154 represents the matrix M given by f(i,j) = max(2i-j, 2j-i) for i >= 1 and j >= 1.  See A204155 for characteristic polynomials of principal submatrices of M, with interlacing zeros. See A204016 for a guide to other choices of M.

From Nathaniel J. Strout, Nov 14 2019: (Start)

The sum of the terms in the n-th "_|" shape is given by the octagonal numbers, A000567. For example,

      5,

      4,

  5,4,3,

  is considered the 3rd such shape.

The sum of the terms in the n-th antidiagonal is the absolute value of the (n+1)-th term of A266085. (End)

LINKS

Robert Israel, Table of n, a(n) for n = 1..10011 (first 141 antidiagonals, flattened)

FORMULA

G.f. as array: (1 + x + y - 7*y*x + 2*y*x^2 + 2*y^2*x)*x*y/((1-x*y)*(1-x)^2*(1-y)^2). - Robert Israel, Dec 03 2017

EXAMPLE

Northwest corner:

  1, 3, 5, 7, 9, ...

  3, 2, 4, 6, 8, ...

  5, 4, 3, 5, 7, ...

  7, 6, 5, 4, 6, ...

  9, 8, 7, 6, 5, ...

  ...

MAPLE

seq(seq(max(3*j-m, 2*m-3*j), j=1..m-1), m=2..19); # Robert Israel, Dec 03 2017

MATHEMATICA

f[i_, j_] := Max[2 i - j, 2 j - i];

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

TableForm[m[8]] (* 8x8 principal submatrix *)

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

  {n, 1, 15}, {i, 1, n}]]  (* A204154 *)

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[%]                 (* A204155 *)

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

CROSSREFS

Cf. A204155, A204016, A202453.

Sequence in context: A142961 A348298 A101777 * A267089 A016555 A214745

Adjacent sequences:  A204151 A204152 A204153 * A204155 A204156 A204157

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Jan 12 2012

STATUS

approved

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Last modified November 29 21:32 EST 2021. Contains 349416 sequences. (Running on oeis4.)