|
| |
|
|
A291939
|
|
First number on layer n of hailstone chain.
|
|
1
|
|
|
|
1, 12, 19, 27, 37, 43, 51, 55, 75, 79, 93, 102, 109, 115, 120, 127, 133, 141, 147, 151, 156, 163, 177, 181, 186, 199, 217, 223, 235, 241, 249, 255, 259, 264, 271, 277, 282, 294, 307, 312, 315, 324, 331, 343, 345, 349, 354, 357, 363, 367
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
COMMENTS
|
If hailstone chains are strictly drawn in numerical order at right angles with consistent direction, overlaps occur. The first set of numbers that do not overlap could be considered the 'first layer'. Once an overlap is needed, all numbers farther up the chain (inclusive) are on a higher layer. This is the sequence of the first numbers to appear on layer n.
The detailed procedure is as follows: Take the "3x+1" or Collatz sequences (CSs) from A070165, and build a 3-dimensional structure representing all CSs up to some starting number (10000). In that 3D structure, the y direction is downwards, x is to the right, and the "layer" z is outside. Process all CSs with increasing starting number coln = 1, 2, 3, ..., 10000. Begin at the end of any individual CS (4, 2, 1), and proceed backwards up to the starting number. Name the elements e[1] = 1, e[2] = 2, e[3] = 4, etc. Position the element e[1] = 1 at coordinates (x,y,z) = (0,0,0). Investigate all e[i] (i > 1): for even e[i] "go right" = store e[i] at (e[i-1].x+1, e[i-1].y, e[i-1].z), for odd e[i] "go down" = store e[i] at (e[i-1].x, e[i-1].y+1, e[i-1].z), whenever that position is not occupied by a different number, otherwise "go out", i.e., increase the layer z by one for this element and all new elements to be stored from now on.
The present sequence A291939 = a(n) consists of the starting values of the CSs which reach a z coordinate of n for the first time.
(End)
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
n=1: the first number is trivially 1;
n=2: in the CS for 12, the 12 overlaps with 13, thus 12 is on layer 2;
n=3: in the CS for 19, the 44 overlaps with 46, thus 19 is on layer 3.
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
