A275122 Pascal's hexagonal pyramid, read by slices, with each slice read by rows.

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 4, 4, 2, 1, 4, 7, 4, 1, 2, 4, 4, 2, 1, 2, 1, 1, 3, 3, 1, 3, 9, 12, 9, 3, 3, 12, 24, 24, 12, 3, 1, 9, 24, 31, 24, 9, 1, 3, 12, 24, 24, 12, 3, 3, 9, 12, 9, 3, 1, 3, 3, 1

Offset: 1

Author

Aresh Pourkavoos, Jul 18 2016

Keywords

• nonn
• look

Comments

Each layer is a hexagon of numbers.

Every cell has 7 neighbors: itself and the 6 around it.

The sum of the values of the neighbors of a cell in one layer is the value of that cell in the next.

Layer 0:

1

Layer 1:

1 1

1 1 1

1 1

Layer 2:

1 2 1

2 4 4 2

1 4 7 4 1

2 4 4 2

1 2 1

Layer 3:

1 3 3 1

.

3 9 12 9 3

.

3 12 24 24 12 3

.

1 9 24 31 24 9 1

.

3 12 24 24 12 3

.

3 9 12 9 3

.

1 3 3 1

etc.

Examples

			Layer 0 is a single 1, so a(1) = 1.
Layer 1 is a filled hexagon of seven 1's, so a(2) through a(8) = 1.
The numbers in the top row of Layer 2, "1 2 1", become terms a(9) through a(11).
		

Links

• Aresh Pourkavoos, Table of n, a(n) for n = 1..10648
• Wikipedia, Hexaflake

Crossrefs

Cf. A046816, A086754.

Programs

Python

import numpy as np
# This is used for the terms[] array
numLayers = 22
# Number of layers that you want to generate
# Number of terms = numLayers^3
width = numLayers*2
# Width and height of the terms[] array
neighbors = [[0, 0], [0, 1], [1, 0], [1, 1], [1, 2], [2, 1], [2, 2]]
# Neighbors of terms that are added together
terms = np.zeros((numLayers, width, width))
# Initialize terms[] array with specified dimensions and fill it with zeros
terms[0][0][0] = 1
# Place a single 1 in layer 0
for l in range(1, numLayers):
  for x in range(width):
    for y in range(width):
      for n in neighbors:
        terms[l][x][y] += terms[l-1][x-n[0]][y-n[1]]
# Calculate terms
seq = terms.flatten().tolist()
# List containing all terms in array
while 0 in seq:
  seq.remove(0)
# Remove zeros from array
for s in range(len(seq)):
  seq[s] = int(seq[s])
# Turn all terms from floats to integers
final = ""
for s in range(len(seq)):
  final += str(s+1)+" "+str(seq[s])+"\n"
# Put the terms into a single string in b-file format
bfile = open("b275122.txt", "w")
bfile.write(final)
bfile.close()
# Write this string to the b-file

Formula

Odd terms in layer x, where x is 1 less than a power of 2, form a hexaflake (conjectured).