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A220073
Mirror of the triangle A130517.
8
1, 1, 2, 2, 1, 3, 3, 1, 2, 4, 4, 2, 1, 3, 5, 5, 3, 1, 2, 4, 6, 6, 4, 2, 1, 3, 5, 7, 7, 5, 3, 1, 2, 4, 6, 8, 8, 6, 4, 2, 1, 3, 5, 7, 9, 9, 7, 5, 3, 1, 2, 4, 6, 8, 10, 10, 8, 6, 4, 2, 1, 3, 5, 7, 9, 11, 11, 9, 7, 5, 3, 1, 2, 4, 6, 8, 10, 12, 12, 10, 8, 6, 4, 2
OFFSET
1,3
COMMENTS
T(n,k) = A130517(n,n-k+1), 1 <= k <= n;
T(n,n) = T(n,1) + 1.
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)
First inverse function (numbers of rows) for pairing function A209293. - Boris Putievskiy, Jan 28 2013
LINKS
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
FORMULA
T(1,1)=1, for n>1: T(n,k)=T(n-1,n-k+1), 1<=k<n and T(n,n)=T(n-1,n)+1.
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| + 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..1..2..3..4..5..6..7...
2..1..1..2..3..4..5..6...
3..2..1..1..2..3..4..5...
4..3..2..1..1..2..3..4...
5..4..3..2..1..1..2..3...
6..5..4..3..2..1..1..2...
7..6..5..4..3..2..1..1...
8..7..6..5..4..3..2..1...
. . .
The start of the sequence as triangle array read by rows:
1,
1, 2,
2, 1, 3,
3, 1, 2, 4,
4, 2, 1, 3, 5,
5, 3, 1, 2, 4, 6,
6, 4, 2, 1, 3, 5, 7,
7, 5, 3, 1, 2, 4, 6, 8,
. . .
Row number r contains r numbers: r-1, r-3,...,1,...r-2,r.
(End)
MATHEMATICA
max = 13;
row[n_] := Join[Range[n, 1, -1], Range[max - n + 1]];
T = Array[row, max];
Table[T[[n - k + 1, k]], {n, 1, max}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Sep 11 2017 *)
PROG
(Haskell)
a220073 n k = a220073_tabl !! (n-1) !! (k-1)
a220073_row n = a220073_tabl !! (n-1)
a220073_tabl = map reverse a130517_tabl
CROSSREFS
Cf. A028310 (left edge), A000027 (right edge), A000012 (central terms), A000217 (row sums), A220075 (partial sums in rows), A002260, A000027, A143182, A187760, A209293.
Sequence in context: A104660 A212125 A282936 * A272020 A264263 A291123
KEYWORD
nonn,tabl
AUTHOR
Reinhard Zumkeller, Dec 03 2012
STATUS
approved