A209304 Table T(n,k)=n+4*k-4 n, k > 0, read by antidiagonals.

1, 5, 2, 9, 6, 3, 13, 10, 7, 4, 17, 14, 11, 8, 5, 21, 18, 15, 12, 9, 6, 25, 22, 19, 16, 13, 10, 7, 29, 26, 23, 20, 17, 14, 11, 8, 33, 30, 27, 24, 21, 18, 15, 12, 9, 37, 34, 31, 28, 25, 22, 19, 16, 13, 10, 41, 38, 35, 32, 29, 26, 23, 20, 17, 14, 11, 45, 42, 39, 36, 33, 30, 27

Offset: 1

Author

Boris Putievskiy, Jan 18 2013

Keywords

• nonn
• tabl

Comments

In general, let m be natural number. The first column of the table T(n,1) is the sequence of the natural numbers A000027. Every next column is formed from previous shifted by m elements.

For m=0 the result is A002260,

for m=1 the result is A002024,

for m=2 the result is A128076,

for m=3 the result is A131914.

This sequence is result for m=4

Examples

			The start of the sequence as table for general case:
  1...m+1...2*m+1...3*m+1...4*m+1...5*m+1...6*m+1  ...
  2...m+2...2*m+2...3*m+2...4*m+2...5*m+2...6*m+2  ...
  3...m+3...2*m+3...3*m+3...4*m+3...5*m+3...6*m+3  ...
  4...m+4...2*m+4...3*m+4...4*m+4...5*m+4...6*m+4  ...
  5...m+5...2*m+5...3*m+5...4*m+5...5*m+5...6*m+5  ...
  6...m+6...2*m+6...3*m+6...4*m+6...5*m+6...6*m+6  ...
  7...m+7...2*m+7...3*m+7...4*m+7...5*m+7...6*m+7  ...
  ...
The start of the sequence as triangle array read by rows for general case:
     1;
    m+1,    2;
  2*m+1,   m+2,   3;
  3*m+1, 2*m+2,   m+3,   4;
  4*m+1, 3*m+2, 2*m+3,   m+4,   5;
  5*m+1, 4*m+2, 3*m+3, 2*m+4,   m+5,  6;
  6*m+1, 5*m+2, 4*m+3, 3*m+4, 2*m+5, m+6, 7;
  ...
Row number r contains r numbers: (r-1)*m+1, (r-2)*m+2,...m+r-1, r.
The start of the sequence as triangle array read by rows for m=4:
  1;
  5,2;
  9,6,3;
  13,10,7,4;
  17,14,11,8,5;
  21,18,15,12,9,6;
  25,22,19,16,13,10,7;
  ...
		

Links

• Boris Putievskiy, Rows n = 1..140 of triangle, flattened
• Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

Crossrefs

Cf. A002260, A002024, A128076, A131914, A003056.

Programs

Python

t=int((math.sqrt(8*n-7) - 1)/ 2)
result = +4*(t+1) + 3*(t*(t+1)/2-n)

Formula

For the general case

a(n) = m*A003056 -(m-1)*A002260.

a(n) = m*(t+1) + (m-1)*(t*(t+1)/2-n),

where t=floor((-1+sqrt(8*n-7))/2).

For m = 4

a(n) = 4*A003056 -3*A002260.

a(n) = 4*(t+1)+3*(t*(t+1)/2-n),

where t=floor((-1+sqrt(8*n-7))/2).