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

1,2

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

Links

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

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

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).

Example

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;
  ...

Prog

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

Crossrefs

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

Sequence in context: A255655 A358516 A177345 * A367844 A258503 A180726

Adjacent sequences: A209301 A209302 A209303 * A209305 A209306 A209307

Keyword

nonn,tabl

Author

Boris Putievskiy, Jan 18 2013

Status

approved