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.
The top left 8x8 corner of the array shows how this surjective self-avoiding walk begins (connect the terms in numerical order, 0-1-2-3-...): +0 +3 +4 +5 58 59 60 63 +1 +2 +7 +6 57 56 61 62 14 13 +8 +9 54 55 50 49 15 12 11 10 53 52 51 48 16 17 30 31 32 33 46 47 19 18 29 28 35 34 45 44 20 23 24 27 36 39 40 43 21 22 25 26 37 38 41 42
b[{n_, k_}, {m_}] := (A[n, k] = m-1);
MapIndexed[b, List @@ HilbertCurve[4][[1]]];
Table[A[n-k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Mar 07 2021 *)
Consider positive integers distributed onto the plane along an increasing Hamiltonian path (in this case it starts downwards):
.
1 4---5---6 59--60--61 64--...
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2---3 8---7 58--57 62--63
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15--14 9--10 55--56 51--50
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16 13--12--11 54--53--52 49
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17--18 31--32--33--34 47--48
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20--19 30--29 36--35 46--45
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21 24--25 28 37 40--41 44
| | | | | | | |
22--23 26--27 38--39 42--43
.
1 is not prime and in adjacent grid cells there are 2 primes: 2 and 3. Therefore a(1) = 2.
2 is prime, therefore a(2) = -1.
8 is not prime and in adjacent grid cells there are 3 primes: 5, 3 and 7. Therefore a(8) = 3.
Replacing n with a(n) in the plane described above, and using "." for a(n) = 0 and "*" for negative a(n), we produce a graph resembling Minesweeper, where the mines are situated at prime n:
2 3---*---3 *---2---* 1 ...
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*---* 3---* 2---2 1---1
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3---3 4---2 3---1 1---.
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2 *---3---* 2---*---2 1
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*---4 *---3---3---2 *---1
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2---* 3---* 2---3 2---2
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2 2---3 2 * 2---* 2
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1---* 1---1 1---2 2---*
In order to produce the sequence, the graph is read along its original mapping.
Block[{nn = 4, s, t, u}, s = ConstantArray[0, {2^#, 2^#}] &[nn + 1]; t = First[HilbertCurve@ # /. Line -> List] &[nn + 1] &[nn + 1]; s = ArrayPad[ReplacePart[s, Array[{1, 1} + t[[#]] -> # &, 2^(2 (nn + 1))]], {{1, 0}, {1, 0}}]; u = Table[If[PrimeQ@ m, -1, Count[#, _?PrimeQ] &@ Union@ Map[s[[#1, #2]] & @@ # &, Join @@ Array[FirstPosition[s, m] + {##} - 2 &, {3, 3}]]], {m, (2^nn)^2}]]
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