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

1,2

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

Table of n, a(n) for n=1..66.

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

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

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

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

Sequence in context: A204890 A307775 A239680 * A076243 A140061 A292776

Adjacent sequences: A128073 A128074 A128075 * A128077 A128078 A128079

Keyword

nonn,tabl,easy

Author

Gary W. Adamson, Feb 14 2007

Extensions

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

Status

approved