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

1,2

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.

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.

Formula

For the general case:

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

As 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).

For b(n) = 2*n-1 and c(n) = 2*n:

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

As 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).

Example

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.

Prog

(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

Crossrefs

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

Sequence in context: A332084 A130266 A261595 * A258246 A258240 A198578

Adjacent sequences: A211194 A211195 A211196 * A211198 A211199 A211200

Keyword

nonn,tabl

Author

Boris Putievskiy, Feb 03 2013

Status

approved