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

1,2

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.

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

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.

Example

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;
  ...

Prog

(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

Crossrefs

Cf. A187760, A004739, A004738.

Sequence in context: A117905 A133445 A139457 * A049992 A227147 A074585

Adjacent sequences: A209298 A209299 A209300 * A209302 A209303 A209304

Keyword

nonn,tabl

Author

Boris Putievskiy, Jan 18 2013

Status

approved