A143182 Triangle T(n,m) = 1 + abs(n-2*m), read by rows, 0<=m<=n.

1, 2, 2, 3, 1, 3, 4, 2, 2, 4, 5, 3, 1, 3, 5, 6, 4, 2, 2, 4, 6, 7, 5, 3, 1, 3, 5, 7, 8, 6, 4, 2, 2, 4, 6, 8, 9, 7, 5, 3, 1, 3, 5, 7, 9, 10, 8, 6, 4, 2, 2, 4, 6, 8, 10, 11, 9, 7, 5, 3, 1, 3, 5, 7, 9, 11, 12, 10, 8, 6, 4, 2, 2, 4, 6, 8, 10, 12, 13, 11, 9, 7, 5, 3, 1, 3, 5, 7, 9, 11, 13

Offset: 0

Author

Roger L. Bagula and Gary W. Adamson, Oct 17 2008

Keywords

• nonn
• easy
• tabl

Comments

From Boris Putievskiy, Jan 15 2013: (Start)

General case see A187760. Let m be natural number. Table T(n,k) n, k > 0, 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. (End)

Examples

			From _Boris Putievskiy_, Jan 15 2013: (Start)
The start of the sequence as table:
1...2...3...4...5...6...7...8...9..10..11...
2...1...2...3...4...5...6...7...8...9..10...
3...2...1...2...3...4...5...6...7...8...9...
4...3...2...1...2...3...4...5...6...7...8...
5...4...3...2...1...2...3...4...5...6...7...
6...5...4...3...2...1...2...3...4...5...6...
7...6...5...4...3...2...1...2...3...4...5...
8...7...6...5...4...3...2...1...2...3...4...
9...8...7...6...5...4...3...2...1...2...3...
10..9...8...7...6...5...4...3...2...1...2...
11.10...9...8...7...6...5...4...3...2...1...
. . .
The start of the sequence as triangle array read by rows: (End)
   1;
   2, 2;
   3, 1, 3;
   4, 2, 2, 4;
   5, 3, 1, 3, 5;
   6, 4, 2, 2, 4, 6;
   7, 5, 3, 1, 3, 5, 7;
   8, 6, 4, 2, 2, 4, 6, 8;
   9, 7, 5, 3, 1, 3, 5, 7, 9;
  10, 8, 6, 4, 2, 2, 4, 6, 8, 10;
  11, 9, 7, 5, 3, 1, 3, 5, 7,  9, 11;
. . .
Row number r contains r numbers: r, r-2,...3,1,3,...r-2,r if r is odd,
r, r-2,...2,2,...r-2,r, if r is even. - _Boris Putievskiy_, Jan 15 2013
		

Links

• G. C. Greubel, Rows n= 0.100 of triangle, flattened
• Boris Putievskiy, Transformations [Of] Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

Crossrefs

Cf. A049581 (subtract 1's), A074148 (row sums), A000027, A220073, A187760.

Programs

GAP

Flat(List([0..15], n-> List([0..n], k-> 1+AbsInt(n-2*k) ))); # G. C. Greubel, Jul 23 2019

Magma

[1+Abs(n-2*k): k in [0..n], n in [0..15]]; // G. C. Greubel, Jul 23 2019

Mathematica

T[n_, m_]:= 1+Abs[(1+n-m) - (1+m)]; Table[Table[t[n, m], {m,0,n}], {n, 0, 15}]//Flatten

PARI

for(n=0,15, for(k=0,n, print1(1+abs(n-2*k), ", "))) \\ G. C. Greubel, Jul 23 2019

Sage

[[1+abs(n-2*k) for k in (0..n)] for n in (0..15)] # G. C. Greubel, Jul 23 2019

Formula

Symmetry: T(n,m) = T(n,n-m).

From Boris Putievskiy, Jan 15 2013: (Start)

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 = 2

a(n) = |(t+1)^2 - 2n| + 2*floor((t^2+3t+2-2n)/(t+1)),

where t=floor((-1+sqrt(8*n-7))/2). (End)

Extensions

Offset and row sums corrected by R. J. Mathar, Jul 05 2012