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

1,2

Links

Boris Putievskiy, Rows n = 1..140 of triangle, flattened

Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

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

Example

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;
  ...

Prog

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

Crossrefs

Cf. A187760.

Sequence in context: A064672 A138554 A063772 * A205122 A174863 A064289

Adjacent sequences: A209299 A209300 A209301 * A209303 A209304 A209305

Keyword

nonn,tabf

Author

Boris Putievskiy, Jan 18 2013

Status

approved