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 3*x if 3*x has not already occurred.
LINKS
Clark Kimberling, Table of n, a(n) for n = 1..1000
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 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Nov 28 2013
STATUS
approved