This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A361374 #35 May 16 2023 14:58:31 %S A361374 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26, %T A361374 27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49, %U A361374 50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,74 %N A361374 Make a square spiral starting with a(1)=1, a(2)=2. Then, each position gets the smallest unused number which is the sum of a path of numbers starting from that position. %C A361374 A path can go in any cardinal direction or diagonal. A path may not repeat the same number. %C A361374 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. %C A361374 a(73) = 72 is the first to break from the natural numbers. 97 is the least positive number which does not occur. %H A361374 Samuel Harkness, <a href="/A361374/b361374.txt">Table of n, a(n) for n = 1..10000</a> %H A361374 Samuel Harkness, <a href="/A361374/a361374.m.txt">MATLAB program</a> %e A361374 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) %e A361374 . %e A361374 37 36 35 34 33 32 31 %e A361374 . %e A361374 38 17 16 15 14 13 30 %e A361374 . %e A361374 39 18 5 4 3 12 29 %e A361374 . %e A361374 40 19 6 1 2 11 28 %e A361374 . %e A361374 41 20 7 8 9 10 27 %e A361374 . / \ %e A361374 start 21 22 23 24 25 26 %e A361374 . %e A361374 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) %e A361374 . %e A361374 65 64 63 62 61 60 59 58 57 %e A361374 . %e A361374 66 37 36 35 34 33 32 31 56 %e A361374 . %e A361374 67 38 17 16 15 14 13 30 55 %e A361374 . %e A361374 68 39 18 5 4 3 12 29 54 %e A361374 . %e A361374 69 40 19 6 1----2 11 28 53 %e A361374 . / %e A361374 70 41 20 7 8 9 10 27 52 %e A361374 . / %e A361374 71 42 21 22 23 24 25 26 51 %e A361374 . / %e A361374 72 43 44 45 46 47 48 49 50 %e A361374 / %e A361374 start %e A361374 . %e A361374 The first 144 terms: %e A361374 . %e A361374 164-162-159-155-153-152-151-149-148-147-146-158 %e A361374 | %e A361374 102-100--99--96--94--93--92--91--90--89-101 154 %e A361374 | | | %e A361374 103 65--64--63--62--61--60--59--58--57 98 150 %e A361374 | | | | | %e A361374 104 66 37--36--35--34--33--32--31 56 95 141 %e A361374 | | | | | | | %e A361374 105 67 38 17--16--15--14--13 30 55 85 140 %e A361374 | | | | | | | | | %e A361374 106 68 39 18 5---4---3 12 29 54 84 139 %e A361374 | | | | | | | | | | | %e A361374 107 69 40 19 6 1---2 11 28 53 82 135 %e A361374 | | | | | | | | | | %e A361374 108 70 41 20 7---8---9--10 27 52 81 134 %e A361374 | | | | | | | | %e A361374 110 71 42 21--22--23--24--25--26 51 88 133 %e A361374 | | | | | | %e A361374 112 72 43--44--45--46--47--48--49--50 87 138 %e A361374 | | | | %e A361374 114 74--73--75--76--77--78--79--80--83--86 137 %e A361374 | | %e A361374 117-116-118-119-120-121-122-124-126-128-132-136 %e A361374 . %e A361374 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. %o A361374 (MATLAB) See Links section. %Y A361374 Cf. A174344, A274923 (spiral coordinates). %Y A361374 Cf. A141481, A334742. %K A361374 nonn %O A361374 1,2 %A A361374 _Samuel Harkness_, Mar 28 2023