OFFSET
1,2
COMMENTS
Let S be the set of numbers defined by these rules: 1 is in S, and if x is in S, then 2*x and 5*x are in S. Then S is the set of positive integers, which arise in generations. Deleting duplicates as they occur, the generations are given by g(1) = (1), g(2) = (2,5), g(3) = (4,10,25), etc. Concatenating these gives A232646, a permutation of the positive integers. For n > 2, the number of numbers in g(n) is n. It is helpful to show the results as a tree with the terms of S as nodes, an edge from x to 2*x if 2*x has not already occurred, and an edge from x to 5*x if 5*x has not already occurred.
LINKS
Aminos Abijad, Rows n = 1..101 of triangle, flattened (rows 1..44 from Clark Kimberling).
FORMULA
Counting the top row as row 0 and writing <i,j> for (2^i)*(5^j), the numbers in row n are <n,0>, <n-1,1>, ..., <0,n>.
EXAMPLE
Each x begets 2*x and 5*x, but if either has already occurred it is deleted. Thus, 1 begets 2 and 5; then 2 begets 4 and 10, and 5 begets only 25, so that g(3) = (4,10,25).
Writing generations as rows results in a triangle whose first five rows are as follows:
1
2 5
4 10 25
8 20 50 125
16 40 100 250 625
...
MATHEMATICA
x = {1}; Do[x = DeleteDuplicates[Flatten[Transpose[{x, 2*x, 5*x}]]], {12}]; x (* Peter J. C. Moses, Nov 27 2013 *)
PROG
(Python)
def row(n): return [2**(n-i)*5**i for i in range(n+1)]
print([e for k in range(10) for e in row(k)]) # Michael S. Branicky, Apr 23 2026
CROSSREFS
KEYWORD
AUTHOR
Clark Kimberling, Nov 28 2013
STATUS
approved