A128076 Triangle T(n,k) = 2*n-k, read by rows.

1, 3, 2, 5, 4, 3, 7, 6, 5, 4, 9, 8, 7, 6, 5, 11, 10, 9, 8, 7, 6, 13, 12, 11, 10, 9, 8, 7, 15, 14, 13, 12, 11, 10, 9, 8, 17, 16, 15, 14, 13, 12, 11, 10, 9, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11

Offset: 1

Author

Gary W. Adamson, Feb 14 2007

Keywords

• nonn
• tabl
• easy

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)

Examples

			First few rows of the triangle are:
1;
3, 2;
5, 4, 3;
7, 6, 5, 4;
9, 8, 7, 6, 5;
...
		

Links

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

Crossrefs

Cf. A128064, A004736, A000326 (row sums), A003056, A002260, A002024, A131914, A209304, A094727 (rows reversed).

Programs

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 *)

Formula

Matrix product A128064 * A004736 as infinite lower triangular matrices.

From Boris Putievskiy, Jan 24 2013: (Start)

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 = 2:

a(n) = 2*A003056 -A002260.

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

a(n) = A105020(n-1)/A002260(n). - Wesley Ivan Hurt, Sep 22 2021

Extensions

NAME simplified. - R. J. Mathar, Sep 27 2021