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A209301 Table T(n,k) n, k > 0, T(n,k)=n-k+1, if n>=k, T(n,k)=k-n+2, if n < k.  Table read by sides of squares from T(1,n) to T(n,n), then from T(n,n) to T(n,1). 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, 9, 8, 7, 6, 5, 4, 3, 1, 2, 3, 4, 5, 6, 7, 8, 10, 9, 8, 7, 6, 5, 4, 3, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 10, 9, 8, 7 (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. 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.

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*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.

EXAMPLE

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;

. . .

PROG

(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

CROSSREFS

Cf.,A187760,A004739,A004738.

Sequence in context: A117905 A133445 A139457 * A049992 A227147 A074585

Adjacent sequences:  A209298 A209299 A209300 * A209302 A209303 A209304

KEYWORD

nonn,tabl

AUTHOR

Boris Putievskiy, Jan 18 2013

STATUS

approved

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Last modified May 29 14:18 EDT 2020. Contains 334700 sequences. (Running on oeis4.)