A209302 Table T(n,k) = max{n+k-1, n+k-1} n, k > 0, read by sides of squares from T(1,n) to T(n,n), then from T(n,n) to T(n,1).

1, 2, 3, 2, 3, 4, 5, 4, 3, 4, 5, 6, 7, 6, 5, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 6, 7, 8, 9, 10, 11, 10, 9, 8, 7, 6, 7, 8, 9, 10, 11, 12, 13, 12, 11, 10, 9, 8, 7, 8, 9, 10, 11, 12, 13, 14, 15, 14, 13, 12, 11, 10, 9, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 16, 15, 14

Offset: 1

Author

Boris Putievskiy, Jan 18 2013

Keywords

• nonn
• tabf

Examples

			The start of the sequence as a table for the general case:
      1   m+1 2*m+1 3*m+1 4*m+1 5*m+1 6*m+1 ...
    m+1   m+2 2*m+2 3*m+2 4*m+2 5*m+2 6*m+2 ...
  2*m+1 2*m+2 2*m+3 3*m+3 4*m+3 5*m+3 6*m+3 ...
  3*m+1 3*m+2 3*m+3 3*m+4 4*m+4 5*m+4 6*m+4 ...
  4*m+1 4*m+2 4*m+3 4*m+4 4*m+5 5*m+5 6*m+5 ...
  5*m+1 5*m+2 5*m+3 5*m+4 5*m+5 5*m+6 6*m+6 ...
  6*m+1 6*m+2 6*m+3 6*m+4 6*m+5 6*m+6 6*m+7 ...
  ...
The start of the sequence as a triangular array read by rows for general case:
      1;
    m+1,   m+2,   m+1;
  2*m+1, 2*m+2, 2*m+3, 2*m+2, 2*m+1;
  3*m+1, 3*m+2, 3*m+3, 3*m+4, 3*m+3, 3*m+2, 3*m+1;
  4*m+1, 4*m+2, 4*m+3, 4*m+4, 4*m+5, 4*m+4, 4*m+3, 4*m+2, 4*m+1;
  ...
Row r contains 2*r-1 terms: r*m+1, r*m+2, ... r*m+r, r*m+r+1, r*m+r, ..., r*m+2, r*m+1.
The start of the sequence as triangle array read by rows for m=1:
  1;
  2,  3,  2;
  3,  4,  5,  4,  3;
  4,  5,  6,  7,  6,  5,  4;
  5,  6,  7,  8,  9,  8,  7,  6,  5;
  6,  7,  8,  9, 10, 11, 10,  9,  8,  7,  6;
  7,  8,  9, 10, 11, 12, 13, 12, 11, 10,  9,  8,  7;
  ...
		

Links

• Boris Putievskiy, Rows n = 1..140 of triangle, flattened
• Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

Crossrefs

Cf. A187760.

Programs

Python

result = 2*int(math.sqrt(n-1)) - abs(n-int(math.sqrt(n-1))**2 - int(math.sqrt(n-1)) -1) +1

Python

from math import isqrt
def A209302(n): return (k:=(m:=isqrt(n))+(n-m*(m+1)>=1))+abs(k**2-n) # Chai Wah Wu, Jun 08 2025

Formula

In general, let m be a natural number. Table T(n,k) = max{m*n+k-m, n+m*k-m}. For the general case,

a(n) = (m+1)*sqrt(n-1) + 1 - |n - floor(sqrt(n-1))^2 - floor(sqrt(n-1))|.

For m=1,

a(n) = 2*sqrt(n-1) + 1 - |n - floor(sqrt(n-1))^2 - floor(sqrt(n-1))|.

a(n) = t + |t^2 - n|, where t = floor(sqrt(n)+1/2). - Ridouane Oudra, May 07 2019