OFFSET
0,2
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)
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.
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)
EXAMPLE
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
MATHEMATICA
T[n_, m_]:= 1+Abs[(1+n-m) - (1+m)]; Table[Table[t[n, m], {m, 0, n}], {n, 0, 15}]//Flatten
PROG
(PARI) for(n=0, 15, for(k=0, n, print1(1+abs(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
(Sage) [[1+abs(n-2*k) for k in (0..n)] for n in (0..15)] # G. C. Greubel, Jul 23 2019
(GAP) Flat(List([0..15], n-> List([0..n], k-> 1+AbsInt(n-2*k) ))); # G. C. Greubel, Jul 23 2019
CROSSREFS
KEYWORD
AUTHOR
Roger L. Bagula and Gary W. Adamson, Oct 17 2008
EXTENSIONS
Offset and row sums corrected by R. J. Mathar, Jul 05 2012
STATUS
approved