A204154 Symmetric matrix based on f(i,j) = max(2i-j, 2j-i), by antidiagonals.

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

Offset: 1

Author

Clark Kimberling, Jan 12 2012

Keywords

• nonn
• tabl

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)

Examples

			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, ...
  ...
		

Links

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

Crossrefs

Cf. A204155, A204016, A202453.

Programs

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}]]

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