1,2

In general, let m be natural number. The first column of the table T(n,1) is the sequence of the natural numbers A000027. In all columns with number k (k > 1) the segment with the length of (k-1): {m+k-2, m+k-3, ..., m} shifts the sequence A000027. For m=1 the result is A004739, for m=2 the result is A004738. This sequence is result for m=3.

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

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

For the general case

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

where

t = floor(sqrt(n)-1/2)+1,

v = floor((n-1)/t)-t+1.

For m=3

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

where

t = floor(sqrt(n)-1/2)+1,

v = floor((n-1)/t)-t+1.

The start of the sequence as table for the general case:

1, m, m+1, m+2, m+3, m+4, m+5, ...

2, 1, m, m+1, m+2, m+3, m+4, ...

3, 2, 1, m, m+1, m+2, m+3, ...

4, 3, 2, 1, m, m+1, m+2, ...

5, 4, 3, 2, 1, m, m+1, ...

6, 5, 4, 3, 2, 1, m, ...

7, 6, 5, 4, 3, 2, 1, ...

...

The start of the sequence as triangle array read by rows for the general case:

1;

m,1,2;

m+1,m,1,2,3;

m+2,m+1,m,1,2,3,4;

m+3,m+2,m+1,m,1,2,3,4,5;

m+4, m+3,m+2,m+1,m,1,2,3,4,5,6;

m+5, m+4, m+3,m+2,m+1,m,1,2,3,4,5,6,7;

...

Row number r contains 2*r -1 numbers: m+r-2, m+r-1,...m,1,2,...r.

The start of the sequence as triangle array read by rows for m=3:

1;

3,1,2;

4,3,1,2,3;

5,4,3,1,2,3,4;

6,5,4,3,1,2,3,4,5;

7,6,5,4,3,1,2,3,4,5,6;

8,7,6,5,4,3,1,2,3,4,5,6,7;

...

(Python)

t=int((math.sqrt(n))-0.5)+1

v=int((n-1)/t)-t+1

result=k*v+(2*v-1)*(t**2-n)+t

Cf. A187760, A004739, A004738.

Sequence in context: A117905 A133445 A139457 * A049992 A227147 A074585

Adjacent sequences: A209298 A209299 A209300 * A209302 A209303 A209304

nonn,tabl

Boris Putievskiy, Jan 18 2013

approved