A259177 Triangle read by rows T(n,k) in which row n lists the even-indexed terms of n-th row of triangle A237593.

1, 2, 1, 2, 1, 3, 2, 3, 1, 1, 4, 2, 1, 4, 2, 1, 5, 2, 2, 5, 2, 1, 1, 6, 3, 1, 1, 6, 2, 1, 2, 7, 3, 1, 2, 7, 3, 2, 1, 8, 3, 1, 1, 2, 8, 3, 1, 1, 2, 9, 4, 1, 1, 2, 9, 3, 2, 1, 2, 10, 4, 2, 1, 2, 10, 4, 1, 2, 2, 11, 4, 1, 1, 1, 3, 11, 4, 2, 1, 1, 2, 12, 5, 2, 1, 1, 2, 12, 4, 2, 1, 1, 3, 13, 5, 1, 1, 2, 3, 13, 5, 2, 1, 2, 2, 14

Offset: 1

Author

Omar E. Pol, Aug 15 2015

Keywords

• nonn
• tabf

Comments

Row n has length A003056(n) hence column k starts in row A000217(k).

Row n is a permutation of the n-th row of A237591 for some n, hence the sequence is a permutation of A237591.

Examples

			Written as an irregular triangle the sequence begins:
1;
2;
1, 2;
1, 3;
2, 3;
1, 1, 4;
2, 1, 4;
2, 1, 5;
2, 2, 5;
2, 1, 1, 6;
3, 1, 1, 6;
2, 1, 2, 7;
3, 1, 2, 7;
3, 2, 1, 8;
3, 1, 1, 2, 8;
3, 1, 1, 2, 9;
...
Illustration of initial terms (side view of the pyramid):
Row                                 _
1                                 _|_|
2                               _|_ _|
3                             _|_|_ _|
4                           _|_|_ _ _|
5                         _|_ _|_ _ _|
6                       _|_|_|_ _ _ _|
7                     _|_ _|_|_ _ _ _|
8                   _|_ _|_|_ _ _ _ _|
9                 _|_ _|_ _|_ _ _ _ _|
10              _|_ _|_|_|_ _ _ _ _ _|
11            _|_ _ _|_|_|_ _ _ _ _ _|
12          _|_ _|_|_ _|_ _ _ _ _ _ _|
13        _|_ _ _|_|_ _|_ _ _ _ _ _ _|
14      _|_ _ _|_ _|_|_ _ _ _ _ _ _ _|
15    _|_ _ _|_|_|_ _|_ _ _ _ _ _ _ _|
16   |_ _ _|_|_|_ _|_ _ _ _ _ _ _ _ _|
...
The above structure represents the first 16 levels (starting from the top) of one of the side views of the infinite stepped pyramid described in A245092. For another side view see A259176.
.
Illustration of initial terms (partial front view of the pyramid):
Row                                 _
1                                  |_|_
2                                 _|_ _|_
3                                |_| |_ _|_
4                               _|_| |_ _ _|_
5                              |_ _|_  |_ _ _|_
6                             _|_| |_| |_ _ _ _|_
7                            |_ _| |_|   |_ _ _ _|_
8                           _|_ _| |_|_  |_ _ _ _ _|_
9                          |_ _|  _|_ _|   |_ _ _ _ _|_
10                        _|_ _| |_| |_|   |_ _ _ _ _ _|_
11                       |_ _ _| |_| |_|_    |_ _ _ _ _ _|_
12                      _|_ _|   |_| |_ _|   |_ _ _ _ _ _ _|_
13                     |_ _ _|  _|_| |_ _|     |_ _ _ _ _ _ _|_
14                    _|_ _ _| |_ _|_  |_|_    |_ _ _ _ _ _ _ _|_
15                   |_ _ _|   |_| |_| |_ _|     |_ _ _ _ _ _ _ _|_
16                   |_ _ _|   |_| |_| |_ _|     |_ _ _ _ _ _ _ _ _|
...
A part of the hidden pattern of the symmetric representation of sigma emerges from the partial front view of the pyramid described in A245092.
For another partial front view see A259176. For the total front view see A237593.
		

Crossrefs

Bisection of A237593.

Row sums give A000027.

Mirror of A259176 which is another bisection of A237593.

Cf. A000203, A000217, A003056, A024916, A175254, A196020, A236104, A237270, A237271, A237591, A244580, A245092, A249351, A259179, A261350.

Programs

Mathematica

(* function f[n,k] and its support functions are defined in A237593 *)
a259177[n_, k_] := f[n, 2*k]
TableForm[Table[a259177[n, k], {n, 1, 16}, {k, 1, row[n]}]] (* triangle *)
Flatten[Table[a259177[n, k], {n, 1, 26}, {k, 1, [n]}]] (* sequence data *)
(* Hartmut F. W. Hoft, Mar 06 2017 *)

PARI

row(n) = (sqrt(8*n + 1) - 1)\2;
s(n, k) = ceil((n + 1)/k - (k + 1)/2) - ceil((n + 1)/(k + 1) - (k + 2)/2);
T(n, k) = if(k<=row(n), s(n, k), s(n, 2*row(n) + 1 - k));
a259177(n, k) = T(n, 2*k);
for(n=1, 26, for(k=1, row(n), print1(a259177(n, k),", ");); print();) \\ Indranil Ghosh, Apr 21 2017

Python

from sympy import sqrt
import math
def row(n): return int(math.floor((sqrt(8*n + 1) - 1)/2))
def s(n, k): return int(math.ceil((n + 1)/k - (k + 1)/2)) - int(math.ceil((n + 1)/(k + 1) - (k + 2)/2))
def T(n, k): return s(n, k) if k<=row(n) else s(n, 2*row(n) + 1 - k)
def a259177(n, k): return T(n, 2*k)
for n in range(1, 27): print([a259177(n, k) for k in range(1, row(n) + 1)]) # Indranil Ghosh, Apr 21 2017

Extensions

Better definition from Omar E. Pol, Apr 26 2021