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A211197
Table T(n,k) = 2*n + ((-1)^n)*(1/2 - (k-1) mod 2) - 1/2; n, k > 0, read by antidiagonals.
1
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, 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
KEYWORD
nonn,tabl
AUTHOR
Boris Putievskiy, Feb 03 2013
STATUS
approved