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], 2012.
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
KEYWORD
nonn,tabl
AUTHOR
Boris Putievskiy, Jan 18 2013
STATUS
approved