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A210530 T(n,k) = (k + 3*n - 2 - (k+n-2)*(-1)^(k+n))/2 n, k > 0, read by antidiagonals. 9
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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 A011448 A174981

Adjacent sequences:  A210527 A210528 A210529 * A210531 A210532 A210533

KEYWORD

nonn,tabl

AUTHOR

Boris Putievskiy, Jan 28 2013

STATUS

approved

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Last modified May 27 02:54 EDT 2019. Contains 323597 sequences. (Running on oeis4.)