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A209304 Table T(n,k)=n+4*k-4 n, k > 0, read by antidiagonals. 3
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 (list; table; graph; refs; listen; history; text; internal format)
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]

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: A306648 A255655 A177345 * A258503 A180726 A018856

Adjacent sequences:  A209301 A209302 A209303 * A209305 A209306 A209307

KEYWORD

nonn,tabl

AUTHOR

Boris Putievskiy, Jan 18 2013

STATUS

approved

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Last modified September 21 16:31 EDT 2021. Contains 347598 sequences. (Running on oeis4.)