A204022 Symmetric matrix based on f(i,j) = max(2i-1, 2j-1), by antidiagonals.

1, 3, 3, 5, 3, 5, 7, 5, 5, 7, 9, 7, 5, 7, 9, 11, 9, 7, 7, 9, 11, 13, 11, 9, 7, 9, 11, 13, 15, 13, 11, 9, 9, 11, 13, 15, 17, 15, 13, 11, 9, 11, 13, 15, 17, 19, 17, 15, 13, 11, 11, 13, 15, 17, 19, 21, 19, 17, 15, 13, 11, 13, 15, 17, 19, 21, 23, 21, 19, 17, 15, 13, 13, 15, 17, 19, 21, 23

Offset: 1

Author

Clark Kimberling, Jan 11 2012

Keywords

• nonn
• tabl

Comments

This sequence represents the matrix M given by f(i,j) = max(2i-1, 2j-1) for i >= 1 and j >= 1. See A204023 for characteristic polynomials of principal submatrices of M, with interlacing zeros. See A204016 for a guide to other choices of M.

Examples

			Northwest corner:
  1 3 5 7 9
  3 3 5 7 9
  5 5 5 7 9
  7 7 7 7 9
  9 9 9 9 9
		

Links

• G. C. Greubel, Rows n = 1..100 of triangle, flattened

Crossrefs

Cf. A202453, A204016, A204022, A209302.

Programs

GAP

Flat(List([1..12], n-> List([1..n], k-> Maximum(2*k-1, 2*(n-k)+1) ))); # G. C. Greubel, Jul 23 2019

Magma

[[Max(2*k-1, 2*(n-k)+1): k in [1..n]]: n in [1..12]]; // G. C. Greubel, Jul 23 2019

Mathematica

(* First program *)
f[i_, j_] := Max[2 i - 1, 2 j - 1];
m[n_] := Table[f[i, j], {i, n}, {j, n}]
TableForm[m[6]] (* 6 X 6 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
  {n, 15}, {i, n}]]                (* A204022 *)
p[n_] := CharacteristicPolynomial[m[n], x];
c[n_] := CoefficientList[p[n], x]
TableForm[Flatten[Table[p[n], {n, 10}]]]
Table[c[n], {n, 12}]
Flatten[%]                         (* A204023 *)
TableForm[Table[c[n], {n, 10}]]
(* Second program *)
Table[Max[2*k-1, 2*(n-k)+1], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Jul 23 2019 *)

PARI

{T(n, k) = max(2*k-1, 2*(n-k)+1)};
for(n=1, 12, for(k=1, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Jul 23 2019

Python

from math import isqrt
def A204022(n): return (m:=isqrt(n<<3)+1>>1)+abs(m**2-(n<<1)+1) # Chai Wah Wu, Jun 08 2025

Sage

[[max(2*k-1, 2*(n-k)+1) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jul 23 2019

Formula

From Ridouane Oudra, May 27 2019: (Start)

a(n) = t + |t^2-2n+1|, where t = floor(sqrt(2n-1)+1/2).

a(n) = A209302(2n-1).

a(n) = A002024(n) + |A002024(n)^2-2n+1|.

a(n) = t + |t^2-2n+1|, where t = floor(sqrt(2n)+1/2). (End)