A187760 Table T(n,k) read by antidiagonals. T(n,k)=n-k+1, if n>=k, T(n,k)=k-n+2, if n < k.

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

Offset

1,2

Comments

In general, let m be natural number. T(n,k)=n-k+1, if n>=k, T(n,k)=k-n+m-1, if n <k. Table T(n,k) read by antidiagonals. The first column of the table T(n,1) is the sequence of the natural numbers A000027. In all columns with number k (k > 1) the segment with the length of (k-1): {m+k-2, m+k-3, ..., m} shifts the sequence A000027. For m=1 the result is A220073, for m=2 the result is A143182. This sequence is the result for m=3.

Links

Boris Putievskiy, Rows n = 1..140 of triangle, flattened

Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

Formula

For the general case, a(n) = |(t+1)^2 - 2n| + m*floor((t^2+3t+2-2n)/(t+1)), where t=floor((-1+sqrt(8*n-7))/2).

For m=3, a(n) = |(t+1)^2 - 2n| + 3*floor((t^2+3t+2-2n)/(t+1)), where t=floor((-1+sqrt(8*n-7))/2).

Example

The start of the sequence as table for the general case:
1....m..m+1..m+2..m+3..m+4..m+5...
2....1....m..m+1..m+2..m+3..m+4...
3....2....1....m..m+1..m+2..m+3...
4....3....2....1....m..m+1..m+2...
5....4....3....2....1....m..m+1...
6....5....4....3....2....1....m...
7....6....5....4....3....2....1...
. . .
The start of the sequence as triangle array read by rows for the general case:
1;
m,2;
m+1,1,3;
m+2,m,2,4;
m+3,m+1,1,3,5;
m+4,m+2,m,2,4,6;
m+5,m+3,m+1,1,3,5,7;
. . .
Row number r contains r numbers: m+r-2, m+r-4,...r-2,r.

Mathematica

T[n_, k_] := If[1 <= k <= n, n - k + 1, k - n + 2];
Table[T[n - k + 1, k], {n, 1, 11}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Nov 06 2018 *)

Prog

(Python)
t=int((math.sqrt(8*n-7)-1)/2)
result=abs((t+1)**2 - 2*n) + 3*int((t**2+3*t+2-2*n)/(t+1))

Crossrefs

Cf. A000027, A220073, A143182.

Sequence in context: A376070 A378423 A121885 * A122143 A144868 A134029

Adjacent sequences: A187757 A187758 A187759 * A187761 A187762 A187763

Keyword

nonn,tabl

Author

Boris Putievskiy, Jan 04 2013

Status

approved