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A338721
Irregular triangle read by rows: T(n,k), n >= 1, k >= 1, in which column k lists the odd numbers k times, and the first element of column k is in row k(k+1)/2.
7
1, 3, 5, 1, 7, 1, 9, 3, 11, 3, 1, 13, 5, 1, 15, 5, 1, 17, 7, 3, 19, 7, 3, 1, 21, 9, 3, 1, 23, 9, 5, 1, 25, 11, 5, 1, 27, 11, 5, 3, 29, 13, 7, 3, 1, 31, 13, 7, 3, 1, 33, 15, 7, 3, 1, 35, 15, 9, 5, 1, 37, 17, 9, 5, 1, 39, 17, 9, 5, 3, 41, 19, 11, 5, 3, 1, 43, 19, 11, 7, 3, 1
OFFSET
1,2
COMMENTS
A missing companion to A196020 and A235791.
T(n,k) is the total number of horizontal steps in the first n levels of the k-th largest double-staircase of the diagram defined in A335616 (see example). - Omar E. Pol, Nov 30 2020
Column k is the partial sums of the k-th column of A339275. - Omar E. Pol, Dec 01 2020
LINKS
FORMULA
T(n,k) = 2 * floor((n-k*(k-1)/2)/k) - 1. - Alois P. Heinz, Nov 30 2020
EXAMPLE
Triangle begins:
1;
3;
5, 1;
7, 1;
9, 3;
11, 3, 1;
13, 5, 1;
15, 5, 1;
17, 7, 3;
19, 7, 3, 1;
21, 9, 3, 1;
23, 9, 5, 1;
25, 11, 5, 1;
27, 11, 5, 3;
29, 13, 7, 3, 1;
31, 13, 7, 3, 1;
33, 15, 7, 3, 1;
35, 15, 9, 5, 1;
37, 17, 9, 5, 1;
39, 17, 9, 5, 3;
41, 19, 11, 5, 3, 1;
43, 19, 11, 7, 3, 1;
45, 21, 11, 7, 3, 1;
47, 21, 13, 7, 3, 1;
49, 23, 13, 7, 5, 1;
51, 23, 13, 9, 5, 1;
53, 25, 15, 9, 5, 3;
55, 25, 15, 9, 5, 3, 1;
...
From Omar E. Pol, Nov 30 2020: (Start)
For an illustration of the rows of triangle consider the infinite "double-staircases" diagram defined in A335616.
For n = 15 the diagram with first 15 levels looks like this:
.
Level "Double-staircases" diagram
. _
1 _|1|_
2 _|1 _ 1|_
3 _|1 |1| 1|_
4 _|1 _| |_ 1|_
5 _|1 |1 _ 1| 1|_
6 _|1 _| |1| |_ 1|_
7 _|1 |1 | | 1| 1|_
8 _|1 _| _| |_ |_ 1|_
9 _|1 |1 |1 _ 1| 1| 1|_
10 _|1 _| | |1| | |_ 1|_
11 _|1 |1 _| | | |_ 1| 1|_
12 _|1 _| |1 | | 1| |_ 1|_
13 _|1 |1 | _| |_ | 1| 1|_
14 _|1 _| _| |1 _ 1| |_ |_ 1|_
15 |1 |1 |1 | |1| | 1| 1| 1|
.
The first largest double-staircase has 29 horizontal steps, the second double-staircase has 13 steps, the third double-staircase has 7 steps, the fourth double-staircase has 3 steps and the fifth double-staircase has only one step, so the 15th row of triangle is [29, 13, 7, 3, 1]. (End)
MAPLE
T:= (n, k)-> 2*iquo(n-k*(k-1)/2, k)-1:
seq(seq(T(n, k), k=1..floor((sqrt(1+8*n)-1)/2)), n=1..30); # Alois P. Heinz, Nov 30 2020
KEYWORD
nonn,tabf
AUTHOR
N. J. A. Sloane, Nov 30 2020.
STATUS
approved