A211161 Table T(n,k) = n, if k is odd, k/2 if k is even; n, k > 0, read by antidiagonals.

1, 1, 2, 1, 1, 3, 2, 2, 1, 4, 1, 2, 3, 1, 5, 3, 2, 2, 4, 1, 6, 1, 3, 3, 2, 5, 1, 7, 4, 2, 3, 4, 2, 6, 1, 8, 1, 4, 3, 3, 5, 2, 7, 1, 9, 5, 2, 4, 4, 3, 6, 2, 8, 1, 10, 1, 5, 3, 4, 5, 3, 7, 2, 9, 1, 11, 6, 2, 5, 4, 4, 6, 3, 8, 2, 10, 1, 12, 1, 6, 3, 5, 5, 4, 7, 3

Offset: 1

Author

Boris Putievskiy, Jan 30 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) = b(n), if k is odd, c(k) if k is even. For this sequence b(n)=n, c(k)=k.

Row T(n,k) is b(n),c(1),b(n),c(2),b(n),c(3),...Numbers c(1),c(2),c(3),... sandwiched between b(n)'s. For this sequence numbers 1,2,3,... (A000027) sandwiched between n's.

Examples

			The start of the sequence as table for general case:
b(1)..c(1)..b(1)..c(2)..b(1)..c(3)..b(1)..c(4)...
b(2)..c(1)..b(2)..c(2)..b(2)..c(3)..b(2)..c(4)...
b(3)..c(1)..b(3)..c(2)..b(3)..c(3)..b(3)..c(4)...
b(4)..c(1)..b(4)..c(2)..b(4)..c(3)..b(4)..c(4)...
b(5)..c(1)..b(5)..c(2)..b(5)..c(3)..b(5)..c(4)...
b(6)..c(1)..b(6)..c(2)..b(6)..c(3)..b(6)..c(4)...
b(7)..c(1)..b(7)..c(2)..b(7)..c(3)..b(7)..c(4)...
b(8)..c(1)..b(8)..c(2)..b(8)..c(3)..b(8)..c(4)...
. . .
The start of the sequence as triangle array read by rows for general case:
b(1);
c(1),b(2);
b(1),c(1),b(3);
c(2),b(2),c(1),b(4);
b(1),c(2),b(3),c(1),b(5);
c(3),b(2),c(2),b(4),c(1),b(6);
b(1),c(3),b(3),c(2),b(5),c(1),b(7);
c(4),b(2),c(3),b(4),c(2),b(6),c(1),b(8);
. . .
Row number r contains r numbers.
If r is odd  b(1),c((r-1)/2),b(3),c((r-1)/2-1),b(5),c((r-1)/2-2),...c(1),b(r).
If r is even c(r/2),b(2),c(r/2-1),b(4),c(r/2-2),b(6),...c(1),b(r).
The start of the sequence as table for b(n)=n and c(k)=k:
1..1..1..2..1..3..1..4...
2..1..2..2..2..3..2..4...
3..1..3..2..3..3..3..4...
4..1..4..2..4..3..4..4...
5..1..5..2..5..3..5..4...
6..1..6..2..6..3..6..4...
7..1..7..2..7..3..7..4...
8..1..8..2..8..3..8..4...
. . .
The start of the sequence as triangle array read by rows for b(n)=n and c(k)=k:
1;
1,2;
1,1,3;
2,2,1,4;
1,2,3,1,5;
3,2,2,4,1,6;
1,3,3,2,5,1,7;
4,2,3,4,2,6,1,8;
. . .
Row number r contains r numbers.
If r is odd  1,(r-1)/2,3,(r-1)/2-1,5,(r-1)/2-2,...1,r.
If r id even r/2,2,r/2-1,4,r/2-1,6,...1,r.
		

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, A002260, A004736, A210530.

Programs

Python

def a(n):
    t=int((math.sqrt(8*n-7) - 1)/ 2)
    i=n-t*(t+1)//2
    j=(t*t+3*t+4)//2-n
    return (1+(-1)**j)*j//4 - (-1+(-1)**j)*i//2

Formula

For the general case

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

As linear sequence

a(n) = (1+(-1)^j)*c(j/2)/2 - (-1+(-1)^j)*b(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) = n and c(k) = k:

As table T(n,k) = (1+(-1)^k)*k/4 - (-1+(-1)^k)*n/2.

As linear sequence a(n) = (1+(-1)^A004736(n))*A004736(n)/4 - (-1+(-1)^A004736(n))*A002260(n)/2. a(n) = (1+(-1)^j)*j/4 - (-1+(-1)^j)*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).