A211197 Table T(n,k) = 2*n + ((-1)^n)*(1/2 - (k-1) mod 2) - 1/2; n, k > 0, read by antidiagonals.

1, 2, 4, 1, 3, 5, 2, 4, 6, 8, 1, 3, 5, 7, 9, 2, 4, 6, 8, 10, 12, 1, 3, 5, 7, 9, 11, 13, 2, 4, 6, 8, 10, 12, 14, 16, 1, 3, 5, 7, 9, 11, 13, 15, 17, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22

Offset: 1

Author

Boris Putievskiy, Feb 03 2013

Keywords

• nonn
• tabl

Comments

In general, let B and C be sequences. By b(n) and c(n) denote elements B and C respectively. Table T(n,k) = (1-(-1)^k)*b(n)/2+(1+(-1)^k)*c(n)/2 read by antidiagonals.

For this sequence b(n)=2*n-1, b(n)=A005408(n), c(n)=2*n, c(n)=A005843(n).

If n is odd row T(n,k) is alternation b(n) and c(n) starts from b(n).

If n is even row T(n,k) is alternation c(n) and b(n) starts from c(n).

For this sequence if n is odd alternation numbers 2*n-1 and 2*n starts from 2*n-1.

For this sequence if n is even alternation numbers 2*n and 2*n-1 starts from 2*n.

T(n,k) is replication of the first and the second columns that are “a braid” from sequences B and C.

Examples

			The start of the sequence as table for general case:
  b(1)..c(1)..b(1)..c(1)..b(1)..c(1)..b(1)..c(1)..
  c(2)..b(2)..c(2)..b(2)..c(2)..b(2)..c(2)..b(2)..
  b(3)..c(3)..b(3)..c(3)..b(3)..c(3)..b(3)..c(3)..
  c(4)..b(4)..c(4)..b(4)..c(4)..b(4)..c(4)..b(4)..
  b(5)..c(5)..b(5)..c(5)..b(5)..c(5)..b(5)..c(5)..
  c(6)..b(6)..c(6)..b(6)..c(6)..b(6)..c(6)..b(6)..
  b(7)..c(7)..b(7)..c(7)..b(7)..c(7)..b(7)..c(7)..
  c(8)..b(8)..c(8)..b(8)..c(8)..b(8)..c(8)..b(8)..
  . . .
The start of the sequence as triangle array read by rows for general case:
  b(1);
  c(1),c(2);
  b(1),b(2),b(3);
  c(1),c(2),c(3),c(4);
  b(1),b(2),b(3),b(4),b(5);
  c(1),c(2),c(3),c(4),c(5),c(6);
  b(1),b(2),b(3),b(4),b(5),b(6),b(7);
  c(1),c(2),c(3),c(4),c(5),c(6),c(7),c(8);
. . .
Row number r contains r numbers.
If r is odd  b(1),b(2),...,b(r).
If r is even c(1),c(2),...,c(r).
The start of the sequence as table for b(n)=2*n-1 and c(n)=2*n:
  1....2...1...2...1...2...1...2...
  4....3...4...3...4...3...4...3...
  5....6...5...6...5...6...5...6...
  8....7...8...7...8...7...8...7...
  9...10...9..10...9..10...9..10...
  12..11..12..11..12..11..12..11...
  13..14..13..14..13..14..13..14...
  16..15..16..15..16..15..16..15...
  . . .
The start of the sequence as triangle array read by rows for  b(n)=2*n-1 and c(n)=2*n:
  1;
  2,4;
  1,3,5;
  2,4,6,8;
  1,3,5,7,9;
  2,4,6,8,10,12;
  1,3,5,7,9,11,13;
  2,4,6,8,10,12,14,16;
  . . .
Row number r contains r numbers.
If r is odd  1,3,...2*r-1 - coincides with the elements row number r triangle array read by rows for sequence 2*A002260-1.
If r is even 2,4,...,2*r  - coincides with the elements row number r triangle array read by rows for sequence 2*A002260.
		

Links

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

Crossrefs

Cf. A000027, A003056, A002260, A004736, A210530, A158405, A005408, A005843, A057211.

Programs

Python

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
result =2*i+((-1)**i)*(0.5 - (j-1) % 2) - 0.5

Python

a211197_list = [2*n - k%2 for k in range(1, 13) for n in range(1, k+1)] # David Radcliffe, Jun 01 2025

Formula

For the general case:

As a table: T(n,k) = (1-(-1)^k)*b(n)/2+(1+(-1)^k)*c(n)/2.

As a linear sequence: a(n) = (1-(-1)^j)*b(i)/2+(1+(-1)^j)*c(i)/2, where i = n-t*(t+1)/2, j = (t*t+3*t+4)/2-n, t = floor((-1+sqrt(8*n-7))/2).

where b(n) = 2*n-1 and c(n) = 2*n.

As a table: T(n,k) = 2*n+((-1)^n)*(1/2- (k-1) mod 2) - 1/2.

As a linear sequence:

a(n) = 2*A002260(n) + ((-1)^A002260(n))*(1/2- (A004736(n)-1) mod 2) -1/2.

a(n) = -(1+(-1)^A003056(n))*A002260(n) +(1+(-1)^A003056(n))*(2*A002260(n)-1)/2.

a(n) = 2*i+((-1)^i)*(1/2- (j-1) mod 2) - 1/2

a(n) = -(1+(-1)^t)*i +(1+(-1)^t)*(2*i-1)/2,

where i = n-t*(t+1)/2, j = (t*t+3*t+4)/2-n, t = floor((-1+sqrt(8*n-7))/2).

T(n, k) = 2*n - 1 + (n+k mod 2); a(n) = 2*A002260(n) - A057211(n). - David Radcliffe, Jun 01 2025