I quite like this chart as a way of thinking about Collatz chains.
Each column after the first one is 2n-1 (where n is the cell to the left). This produces the pattern given in the column titles, 2n+1, 4n+1 etc.
Numbers in the 2n+1 column produce upward moves within the column (eg 15 -> 23 -> 35) or a move to a number to the right of the column. Numbers in the 8n+1 column and all columns to the right produce downward moves, two columns to the left and down the chart (eg 65 -> 49 -> 37).
The 4n+1 contains all gateway numbers (1, 5, 21, 85) which lead to a power of 2 and thus to 1. Other numbers in this column can lead to any other column or a move within the column.
