A209301 Table T(n,k) n, k > 0, T(n,k)=n-k+1, if n>=k, T(n,k)=k-n+2, if n < k. Table read by sides of squares from T(1,n) to T(n,n), then from T(n,n) to T(n,1).

1, 3, 1, 2, 4, 3, 1, 2, 3, 5, 4, 3, 1, 2, 3, 4, 6, 5, 4, 3, 1, 2, 3, 4, 5, 7, 6, 5, 4, 3, 1, 2, 3, 4, 5, 6, 8, 7, 6, 5, 4, 3, 1, 2, 3, 4, 5, 6, 7, 9, 8, 7, 6, 5, 4, 3, 1, 2, 3, 4, 5, 6, 7, 8, 10, 9, 8, 7, 6, 5, 4, 3, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 10, 9, 8, 7

Offset: 1

Author

Boris Putievskiy, Jan 18 2013

Keywords

• nonn
• tabl

Comments

In general, let m be natural number. The first column of the table T(n,1) is the sequence of the natural numbers A000027. In all columns with number k (k > 1) the segment with the length of (k-1): {m+k-2, m+k-3, ..., m} shifts the sequence A000027. For m=1 the result is A004739, for m=2 the result is A004738. This sequence is result for m=3.

Examples

			The start of the sequence as table for the general case:
  1,   m, m+1, m+2, m+3, m+4, m+5, ...
  2,   1,   m, m+1, m+2, m+3, m+4, ...
  3,   2,   1,   m, m+1, m+2, m+3, ...
  4,   3,   2,   1,   m, m+1, m+2, ...
  5,   4,   3,   2,   1,   m, m+1, ...
  6,   5,   4,   3,   2,   1,   m, ...
  7,   6,   5,   4,   3,   2,   1, ...
  ...
The start of the sequence as triangle array read by rows for the general case:
  1;
  m,1,2;
  m+1,m,1,2,3;
  m+2,m+1,m,1,2,3,4;
  m+3,m+2,m+1,m,1,2,3,4,5;
  m+4, m+3,m+2,m+1,m,1,2,3,4,5,6;
  m+5, m+4, m+3,m+2,m+1,m,1,2,3,4,5,6,7;
  ...
Row number r contains 2*r -1 numbers: m+r-2, m+r-1,...m,1,2,...r.
The start of the sequence as triangle array read by rows for m=3:
  1;
  3,1,2;
  4,3,1,2,3;
  5,4,3,1,2,3,4;
  6,5,4,3,1,2,3,4,5;
  7,6,5,4,3,1,2,3,4,5,6;
  8,7,6,5,4,3,1,2,3,4,5,6,7;
  ...
		

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, A004739, A004738.

Programs

Python

t=int((math.sqrt(n))-0.5)+1
v=int((n-1)/t)-t+1
result=k*v+(2*v-1)*(t**2-n)+t

Formula

For the general case

a(n) = m*v+(2*v-1)*(t*t-n)+t,

where

t = floor(sqrt(n)-1/2)+1,

v = floor((n-1)/t)-t+1.

For m=3

a(n) = 3*v+(2*v-1)*(t*t-n)+t,

where

t = floor(sqrt(n)-1/2)+1,

v = floor((n-1)/t)-t+1.