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A211161 Table T(n,k) = n, if k is odd, k/2 if k is even; n, k > 0, read by antidiagonals. 2
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 (list; table; graph; refs; listen; history; text; internal format)
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 * A208101 A131333 A120643

Adjacent sequences:  A211158 A211159 A211160 * A211162 A211163 A211164

KEYWORD

nonn,tabl

AUTHOR

Boris Putievskiy, Jan 30 2013

STATUS

approved

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Last modified August 1 22:36 EDT 2021. Contains 346408 sequences. (Running on oeis4.)