A204164 Symmetric matrix based on f(i,j) = floor((i+j)/2), by antidiagonals.

1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 1

Author

Clark Kimberling, Jan 12 2012

Keywords

• nonn
• easy
• tabl

Comments

A204164 represents the matrix M given by f(i,j) = floor((i+j)/2) for i >= 1 and j >= 1. See A204165 for characteristic polynomials of principal submatrices of M, with interlacing zeros. See A204016 for a guide to other choices of M.

k appears 4k-1 times, k > 0. - Boris Putievskiy, Jun 12 2024

Number of numbers of the form 2k^2+k+1 <= n, for k = 0,1,2,... - Wesley Ivan Hurt, Jun 19 2024

Examples

			Northwest corner:
  1 1 2 2 3 3 4 4
  1 2 2 3 3 4 4 5
  2 2 3 3 4 4 5 5
  2 3 3 4 4 5 5 6
  3 3 4 4 5 5 6 6
		

Links

• Boris Putievskiy, Table of n, a(n) for n = 1..9870
• Boris Putievskiy, Integer Sequences: Irregular Arrays and Intra-Block Permutations, arXiv:2310.18466 [math.CO], 2023.

Crossrefs

Cf. A057211, A204165, A204016, A202453, A370655.

Programs

Mathematica

f[i_, j_] := Floor[(i + j)/2];
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}]]  (* this sequence *)
  (* or *)
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[%]                 (* A204165 *)
TableForm[Table[c[n], {n, 1, 10}]]
  (* or *)
a[n_] = Ceiling[(Sqrt[8*n + 1] - 1)/4];
Nmax = 21; Table[a[n], {n, 1, Nmax}] (* Boris Putievskiy, Jun 12 2024 *)

Python

from math import isqrt
def A204164(n): return (m:=isqrt(n>>1))+(n>m*((m<<1)+1)) # Chai Wah Wu, Nov 14 2024

Formula

a(n) = ceiling((sqrt(8*n+1)-1)/4). - Boris Putievskiy, Jun 12 2024

a(n) = Sum_{k=1..n} [c(k) = c(k-1)+1], where c(n) = floor(sqrt(2n)+1/2) mod 2 = A057211(n) and [] is the Iverson bracket. - Wesley Ivan Hurt, Jun 23 2024

a(n) = m+1 if n>m(2m+1) and a(n) = m otherwise where m = floor(sqrt(n/2)). - Chai Wah Wu, Nov 14 2024