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
KEYWORD
nonn,tabl
AUTHOR
Boris Putievskiy, Jan 30 2013
STATUS
approved