OFFSET
1,2
COMMENTS
A path can go in any cardinal direction or diagonal. A path may not repeat the same number.
For a while, this sequence seems to simply be the natural numbers. However, the percentage of natural numbers in this sequence tends to 0. E.g., only 2347 of the first million natural numbers are in this sequence.
a(73) = 72 is the first to break from the natural numbers. 97 is the least positive number which does not occur.
LINKS
Samuel Harkness, Table of n, a(n) for n = 1..10000
Samuel Harkness, MATLAB program
EXAMPLE
For a(42), the first candidate to check is 42, as it is the least unused positive integer. 20-22 is a valid path which ends at a(42) and whose sum is 42, so a(42) = 42. (path shown below)
.
37 36 35 34 33 32 31
.
38 17 16 15 14 13 30
.
39 18 5 4 3 12 29
.
40 19 6 1 2 11 28
.
41 20 7 8 9 10 27
. / \
start 21 22 23 24 25 26
.
For a(73), the first candidate to check is 73, as it is the least unused positive integer. No paths starting at a(73) equal 73, so check the next candidate, 74. 43-21-7-1-2 is a valid path starting at a(73) and whose sum is 74, so a(73) = 74. (path shown below)
.
65 64 63 62 61 60 59 58 57
.
66 37 36 35 34 33 32 31 56
.
67 38 17 16 15 14 13 30 55
.
68 39 18 5 4 3 12 29 54
.
69 40 19 6 1----2 11 28 53
. /
70 41 20 7 8 9 10 27 52
. /
71 42 21 22 23 24 25 26 51
. /
72 43 44 45 46 47 48 49 50
/
start
.
The first 144 terms:
.
164-162-159-155-153-152-151-149-148-147-146-158
|
102-100--99--96--94--93--92--91--90--89-101 154
| | |
103 65--64--63--62--61--60--59--58--57 98 150
| | | | |
104 66 37--36--35--34--33--32--31 56 95 141
| | | | | | |
105 67 38 17--16--15--14--13 30 55 85 140
| | | | | | | | |
106 68 39 18 5---4---3 12 29 54 84 139
| | | | | | | | | | |
107 69 40 19 6 1---2 11 28 53 82 135
| | | | | | | | | |
108 70 41 20 7---8---9--10 27 52 81 134
| | | | | | | |
110 71 42 21--22--23--24--25--26 51 88 133
| | | | | |
112 72 43--44--45--46--47--48--49--50 87 138
| | | |
114 74--73--75--76--77--78--79--80--83--86 137
| |
117-116-118-119-120-121-122-124-126-128-132-136
.
Note that 97 does not (and will not) occur. A path must start with one of the outer-most cells, all of which are greater than 97, and nothing below their minimum can ever be reached again.
PROG
(MATLAB) See Links section.
CROSSREFS
KEYWORD
nonn
AUTHOR
Samuel Harkness, Mar 28 2023
STATUS
approved