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A143182
Triangle T(n,m) = 1 + abs(n-2*m), read by rows, 0<=m<=n.
11
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
0,2
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
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
Cf. A049581 (subtract 1's), A074148 (row sums), A000027, A220073, A187760.
Sequence in context: A204123 A237448 A204143 * A128715 A237447 A368053
KEYWORD
nonn,easy,tabl
AUTHOR
EXTENSIONS
Offset and row sums corrected by R. J. Mathar, Jul 05 2012
STATUS
approved