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

%I A275122 #28 Oct 14 2019 01:28:23
%S A275122 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,
%T A275122 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,
%U A275122 3,1
%N A275122 Pascal's hexagonal pyramid, read by slices, with each slice read by rows.
%C A275122 Each layer is a hexagon of numbers.
%C A275122 Every cell has 7 neighbors: itself and the 6 around it.
%C A275122 The sum of the values of the neighbors of a cell in one layer is the value of that cell in the next.
%C A275122 Layer 0:
%C A275122   1
%C A275122 Layer 1:
%C A275122    1 1
%C A275122   1 1 1
%C A275122    1 1
%C A275122 Layer 2:
%C A275122     1 2 1
%C A275122    2 4 4 2
%C A275122   1 4 7 4 1
%C A275122    2 4 4 2
%C A275122     1 2 1
%C A275122 Layer 3:
%C A275122         1   3   3   1
%C A275122 .
%C A275122       3   9  12   9   3
%C A275122 .
%C A275122     3  12  24  24  12   3
%C A275122 .
%C A275122   1   9  24  31  24   9   1
%C A275122 .
%C A275122     3  12  24  24  12   3
%C A275122 .
%C A275122       3   9  12   9   3
%C A275122 .
%C A275122         1   3   3   1
%C A275122 etc.
%H A275122 Aresh Pourkavoos, <a href="/A275122/b275122.txt">Table of n, a(n) for n = 1..10648</a>
%H A275122 Wikipedia, <a href="https://en.wikipedia.org/wiki/N-flake#Hexaflake">Hexaflake</a>
%F A275122 Odd terms in layer x, where x is 1 less than a power of 2, form a hexaflake (conjectured).
%e A275122 Layer 0 is a single 1, so a(1) = 1.
%e A275122 Layer 1 is a filled hexagon of seven 1's, so a(2) through a(8) = 1.
%e A275122 The numbers in the top row of Layer 2, "1 2 1", become terms a(9) through a(11).
%o A275122 (Python)
%o A275122 import numpy as np
%o A275122 # This is used for the terms[] array
%o A275122 numLayers = 22
%o A275122 # Number of layers that you want to generate
%o A275122 # Number of terms = numLayers^3
%o A275122 width = numLayers*2
%o A275122 # Width and height of the terms[] array
%o A275122 neighbors = [[0, 0], [0, 1], [1, 0], [1, 1], [1, 2], [2, 1], [2, 2]]
%o A275122 # Neighbors of terms that are added together
%o A275122 terms = np.zeros((numLayers, width, width))
%o A275122 # Initialize terms[] array with specified dimensions and fill it with zeros
%o A275122 terms[0][0][0] = 1
%o A275122 # Place a single 1 in layer 0
%o A275122 for l in range(1, numLayers):
%o A275122   for x in range(width):
%o A275122     for y in range(width):
%o A275122       for n in neighbors:
%o A275122         terms[l][x][y] += terms[l-1][x-n[0]][y-n[1]]
%o A275122 # Calculate terms
%o A275122 seq = terms.flatten().tolist()
%o A275122 # List containing all terms in array
%o A275122 while 0 in seq:
%o A275122   seq.remove(0)
%o A275122 # Remove zeros from array
%o A275122 for s in range(len(seq)):
%o A275122   seq[s] = int(seq[s])
%o A275122 # Turn all terms from floats to integers
%o A275122 final = ""
%o A275122 for s in range(len(seq)):
%o A275122   final += str(s+1)+" "+str(seq[s])+"\n"
%o A275122 # Put the terms into a single string in b-file format
%o A275122 bfile = open("b275122.txt", "w")
%o A275122 bfile.write(final)
%o A275122 bfile.close()
%o A275122 # Write this string to the b-file
%Y A275122 Cf. A046816, A086754.
%K A275122 nonn,look
%O A275122 1,10
%A A275122 _Aresh Pourkavoos_, Jul 18 2016