OFFSET
1,2
COMMENTS
From Boris Putievskiy, Jan 24 2013: (Start)
Table T(n,k) = n+2*k-2 n, k > 0, read by antidiagonals.
General case A209304. 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,
for m=4 the result is A209304. (End)
LINKS
Boris Putievskiy, Transformations [Of] Integer Sequences And Pairing Functions, arXiv preprint arXiv:1212.2732 [math.CO], 2012.
FORMULA
From Boris Putievskiy, Jan 24 2013: (Start)
For the general case:
a(n) = m*(t+1) + (m-1)*(t*(t+1)/2-n), where t=floor((-1+sqrt(8*n-7))/2).
For m = 2:
a(n) = 2*(t+1)+(t*(t+1)/2-n), where t=floor((-1+sqrt(8*n-7))/2). (End)
a(n) = (r^2 + 3*r - 2*n)/2, where r = round(sqrt(2*n)). - Wesley Ivan Hurt, Sep 19 2021
EXAMPLE
First few rows of the triangle are:
1;
3, 2;
5, 4, 3;
7, 6, 5, 4;
9, 8, 7, 6, 5;
...
MAPLE
A128076 := proc(n, k)
2*n-k ;
end proc:
seq(seq( A128076(n, k), k=1..n), n=1..12) ; # R. J. Mathar, Sep 27 2021
MATHEMATICA
Table[(Round[Sqrt[2 n]]^2 + 3 Round[Sqrt[2 n]] - 2 n)/2, {n, 100}] (* Wesley Ivan Hurt, Sep 19 2021 *)
CROSSREFS
KEYWORD
AUTHOR
Gary W. Adamson, Feb 14 2007
EXTENSIONS
NAME simplified. - R. J. Mathar, Sep 27 2021
STATUS
approved