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