OFFSET
1,2
COMMENTS
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.
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
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Aug 15 2015
EXTENSIONS
Better definition from Omar E. Pol, Apr 26 2021
STATUS
approved