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

1,3

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.

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)*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).

Example

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.

Prog

(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

Crossrefs

Cf. A000027, A002260, A004736, A210530.

Sequence in context: A345116 A278042 A338714 * A348687 A208101 A131333

Adjacent sequences: A211158 A211159 A211160 * A211162 A211163 A211164

Keyword

nonn,tabl

Author

Boris Putievskiy, Jan 30 2013

Status

approved