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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. 7
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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: A140430 A123359 A121885 * A122143 A144868 A134029

Adjacent sequences:  A187757 A187758 A187759 * A187761 A187762 A187763

KEYWORD

nonn,tabl

AUTHOR

Boris Putievskiy, Jan 04 2013

STATUS

approved

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Last modified February 19 11:15 EST 2019. Contains 320310 sequences. (Running on oeis4.)