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A259176 Triangle read by rows T(n,k) which is a bisection of A237593. 12
1, 2, 2, 1, 3, 1, 3, 2, 4, 1, 1, 4, 1, 2, 5, 1, 2, 5, 2, 2, 6, 1, 1, 2, 6, 1, 1, 3, 7, 2, 1, 2, 7, 2, 1, 3, 8, 1, 2, 3, 8, 2, 1, 1, 3, 9, 2, 1, 1, 3, 9, 2, 1, 1, 4, 10, 2, 1, 2, 3, 10, 2, 1, 2, 4, 11, 2, 2, 1, 4, 11, 3, 1, 1, 1, 4, 12, 2, 1, 1, 2, 4, 12, 2, 1, 1, 2, 5, 13, 3, 1, 1, 2, 4, 13, 3, 2, 1, 1, 5, 14, 2, 2, 1, 2, 5 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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.

Row sums give A000027.

For the mirror see A259177, another bisection of A237593.

LINKS

Table of n, a(n) for n=1..106.

EXAMPLE

Written as an irregular triangle the sequence begins:

1;

2;

2, 1;

3, 1;

3, 2;

4, 1, 1;

4, 1, 2;

5, 1, 2;

5, 2, 2;

6, 1, 1, 2;

6, 1, 1, 3;

7, 2, 1, 2;

7, 2, 1, 3;

8, 1, 2, 3;

8, 2, 1, 1, 3;

9, 2, 1, 1, 3;

...

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 A259177.

.

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 A259177. For the total front view see A237593.

MATHEMATICA

(* function f[n, k] and its support functions are defined in A237593 *)

a259176[n_, k_] := f[n, 2*k-1]

TableForm[Table[a259176[n, k], {n, 1, 16}, {k, 1, row[n]}]] (* triangle *)

Flatten[Table[a259176[n, k], {n, 1, 26}, {k, 1, [n]}]] (* sequence data *)

(* Hartmut F. W. Hoft, Mar 06 2017 *)

PROG

(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 - 1);

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 - 1)

for n in range(1, 11): print([a259177(n, k) for k in range(1, row(n) + 1)]) # Indranil Ghosh, Apr 21 2017

CROSSREFS

Cf. A000203, A000217, A003056, A024916, A175254, A196020, A236104, A237270, A237271, A237591, A237593, A244580, A245092, A249351, A259177, A259179, A261350.

Sequence in context: A175245 A167413 A340985 * A237591 A277730 A174167

Adjacent sequences:  A259173 A259174 A259175 * A259177 A259178 A259179

KEYWORD

nonn,tabf

AUTHOR

Omar E. Pol, Aug 15 2015

STATUS

approved

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Last modified April 15 00:13 EDT 2021. Contains 342971 sequences. (Running on oeis4.)