A210530 T(n,k) = (k + 3*n - 2 - (k+n-2)*(-1)^(k+n))/2 n, k > 0, read by antidiagonals.

1, 2, 3, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22

Offset

1,2

Comments

Row T(n,k) for odd n is even numbers sandwiched between n's starts from n and 2*n.

Row T(n,k) for even n is odd numbers sandwiched between n's starts from 2*n-1 and n.

Antidiagonal T(1,k), T(2,k-1), ..., T(k,1) for odd k is 1,2,3,...,k.

Antidiagonal T(1,k), T(2,k-1), ..., T(k,1) for even k is k+1, k+2, ..., 2*k+1.

The main diagonal is A000027.

Diagonal, located above the main diagonal T(1,k), T(2,k+1), T(3,k+2), ... for odd k is A000027.

Diagonal, located above the main diagonal T(1,k), T(2,k+1), T(3,k+2), ... for even k is k, k+3, k+6, ..., A016789, A016777, A008585.

Diagonal, located below the main diagonal T(n,1), T(n+1,2), T(n+2,3), ... for odd n is n,n+1, n+2, ... A000027.

Diagonal, located below the main diagonal T(n,1), T(n+1,2), T(n+2,3), ... for even n is 2*n-1, 2*n+2, 2*n+5, ... A008585, A016777, A016789.

The table contains:

A124625 as row 1,

A114753 as column 1,

A109043 as column 2,

A066104 as column 4.

Links

Boris Putievskiy, Rows n = 1..140 of triangle, flattened

Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

Formula

As table T(n,k) = (k + 3*n - 2 - (k+n-2)*(-1)^(k+n))/2.

As linear sequence

a(n) = A000027(n) - A204164(n)*(2*A204164(n)-3) - 1.

a(n) = n - v*(2*v-3) - 1, where t = floor((-1 + sqrt(8*n-7))/2) and v = floor((t+2)/2).

G.f. of the table: (y*(- 1 + 3*y^2) + x^2*(2 + 5*y - 2*y^2 - 7*y^3) + x^3*(4 + y - 6*y^2 - y^3) + x*(y + 2*y^2 - y^3))/((- 1 + x)^2*(1 + x)^2*(-1 + y)^2*(1 + y)^2). - Stefano Spezia, Nov 17 2018

Example

The start of the sequence as table:
   1   2   1   4   1   6   1   8   1  10
   3   2   5   2   7   2   9   2  11   2
   3   6   3   8   3  10   3  12   3  14
   7   4   9   4  11   4  13   4  15   4
   5  10   5  12   5  14   5  16   5  18
  11   6  13   6  15   6  17   6  19   6
   7  14   7  16   7  18   7  20   7  22
  15   8  17   8  19   8  21   8  23   8
   9  18   9  20   9  22   9  24   9  26
  19  10  21  10  23  10  25  10  27  10
  ...
The start of the sequence as triangle array read by rows:
   1;
   2,  3;
   1,  2,  3;
   4,  5,  6,  7;
   1,  2,  3,  4,  5;
   6,  7,  8,  9, 10, 11;
   1,  2,  3,  4,  5,  6,  7;
   8,  9, 10, 11, 12, 13, 14, 15;
   1,  2,  3,  4,  5,  6,  7,  8,  9;
  10, 11, 12, 13, 14, 15, 16, 17, 18, 19;
  ...
Row number r contains r numbers.
If r is  odd: 1,2,3,...,r.
If r is even: r, r+1, r+3, ..., 2*r-1.
The start of the sequence as array read by rows, the length of row r is 4*r-1.
First 2*r-1 numbers are from the row number 2*r-1 of triangle array, located above.
Last 2*r numbers are from the row number 2*r of triangle array, located above.
  1,2,3;
  1,2,3,4,5,6,7;
  1,2,3,4,5,6,7,8,9,10,11;
  1,2,3,4,5,6,7,8,9,10,11,12,13,14,15;
  1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19;
  ...
Row number r contains 4*r-1 numbers: 1,2,3,...,4*r-1.

Mathematica

T[n_, k_] := (k+3n-2-(k+n-2)(-1)^(k+n))/2; Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Nov 17 2018 *)

Prog

(PARI) T(n, k) = (k+3*n-2-(k+n-2)*(-1)^(k+n))/2; \\ Andrew Howroyd, Jan 11 2018
(Python)
t=int((math.sqrt(8*n-7)-1)/2)
v=int((t+2)/2)
result=n-v*(2*v-3)-1

Crossrefs

Cf. A124625, A114753, A109043, A066104, A000027, A016789, A016777, A008585, A204164.

Sequence in context: A025481 A124171 A276146 * A076645 A350604 A011448

Adjacent sequences: A210527 A210528 A210529 * A210531 A210532 A210533

Keyword

nonn,tabl

Author

Boris Putievskiy, Jan 28 2013

Status

approved