A232646 Sequence (or tree or triangle) generated by these rules: 1 is in S, and if x is in S, then 2*x and 5*x + 3 are in S, and duplicates are deleted as they occur.

1, 2, 5, 4, 10, 25, 8, 20, 50, 125, 16, 40, 100, 250, 625, 32, 80, 200, 500, 1250, 3125, 64, 160, 400, 1000, 2500, 6250, 15625, 128, 320, 800, 2000, 5000, 12500, 31250, 78125, 256, 640, 1600, 4000, 10000, 25000, 62500, 156250, 390625, 512, 1280, 3200, 8000

Offset: 1

Author

Clark Kimberling, Nov 28 2013

Keywords

• nonn
• easy

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.

Examples

			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
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A232559, A232639, A036561.

Programs

Mathematica

x = {1}; Do[x = DeleteDuplicates[Flatten[Transpose[{x, 2*x, 5*x}]]], {12}]; x  (* Peter J. C. Moses, Nov 27 2013 *)

Formula

Counting the top row as row 0 and writing for (2^i)*(5*j) , the numbers in row n are , , ..., <0,n>.