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 A359005 #47 Jun 24 2023 16:03:02
%S A359005 0,1,2,4,7,3,5,8,13,6,10,16,25,12,19,9,14,22,34,52,79,39,59,29,44,21,
%T A359005 32,15,23,11,17,26,40,61,30,46,70,106,160,241,120,181,90,136,67,33,50,
%U A359005 24,37,18,28,43,65,98,48,73,36,55,27,41,20,31,47,71,35,53
%N A359005 Jane Street's infinite sidewalk's greedy walk.
%C A359005 The Jane Street greedy walk is obtained by starting on slab 0 of Jane Street's infinite sidewalk (see A358838) and always jumping backwards preferably than forwards. Also, it is forbidden to visit a slab more than once.
%C A359005 The sequence is well defined. It demonstrably never ends and never repeats.
%C A359005 It is conjectured that the sequence actually visits all nonnegative integers. This is related to the conjecture that given any positive integer n, there exists a minimal Jane Street's path (i.e., with minimal number of jumps) going from slab 0 to slab n, and ending either in "forwards" or in "forwards-forwards-backwards". This would make the sequence a one-to-one mapping of the nonnegative numbers onto themselves. See also A359008.
%H A359005 Neal Gersh Tolunsky, <a href="/A359005/b359005.txt">Table of n, a(n) for n = 0..10000</a>
%e A359005 Let N denote the nonnegative integers.
%e A359005 Let forward and backward denote the forward-jumping and backward-jumping functions as defined in Jane Street's infinite sidewalk rules (see A358838):
%e A359005 forward(n) = n + floor(n/2) + 1
%e A359005 backward(n) = n - floor(n/2) - 1
%e A359005 Finally, let f denote the (demonstrably one-to-one) mapping of N x N to N:
%e A359005 f(i, j) = forward^j(3*i)
%e A359005 The mapping f is illustrated by the following table, where the contents of each cell (i, j) is f(i, j):
%e A359005 .
%e A359005 \j|
%e A359005 i\| 0 1 2 3 4 5 6 7 8 ...
%e A359005 ---+------------------------------------------------
%e A359005 0 | 0 1 2 4 7 11 17 26 40 ...
%e A359005 1 | 3 5 8 13 20 31 47 71 107 ...
%e A359005 2 | 6 10 16 25 38 58 88 133 200 ...
%e A359005 3 | 9 14 22 34 52 79 119 179 269 ...
%e A359005 4 | 12 19 29 44 67 101 152 229 344 ...
%e A359005 5 | 15 23 35 53 80 121 182 274 412 ...
%e A359005 6 | 18 28 43 65 98 148 223 335 503 ...
%e A359005 7 | 21 32 49 74 112 169 254 382 574 ...
%e A359005 8 | 24 37 56 85 128 193 290 436 655 ...
%e A359005 9 | 27 41 62 94 142 214 322 484 727 ...
%e A359005 10 | 30 46 70 106 160 241 362 544 817 ...
%e A359005 11 | 33 50 76 115 173 260 391 587 881 ...
%e A359005 12 | 36 55 83 125 188 283 425 638 958 ...
%e A359005 13 | 39 59 89 134 202 304 457 686 1030 ...
%e A359005 ...| ... ... ... ... ... ... ... ... ... ...
%e A359005 (Note that the first row of the table corresponds to A006999.)
%e A359005 .
%e A359005 Equivalently, we can represent f(i, j) as a(n) for some n. This results in the following alternate illustrative table, where the contents of each cell (i, j) is still f(i, j), but represented as a(n):
%e A359005 .
%e A359005 \j|
%e A359005 i\| 0 1 2 3 4 5 6 7 8 ...
%e A359005 ---+------------------------------------------------------------------------
%e A359005 0 | a(0) a(1) a(2) a(3) a(4) a(29) a(30) a(31) a(32)
%e A359005 1 | a(5) a(6) a(7) a(8) a(60) a(61) a(62) a(63) a(91)
%e A359005 2 | a(9) a(10) a(11) a(12) a(75) a(76) a(77) a(78) a(412)
%e A359005 3 | a(15) a(16) a(17) a(18) a(19) a(20) a(297) a(298) a(1109)
%e A359005 4 | a(13) a(14) a(23) a(24) a(44) a(99) a(100) a(258) a(435)
%e A359005 5 | a(27) a(28) a(64) a(65) a(66) a(67) a(206) a(207) a(208)
%e A359005 6 | a(49) a(50) a(51) a(52) a(53) a(284) a(285) a(286) a(287)
%e A359005 7 | a(25) a(26) a(81) a(82) a(83) a(84) a(418) a(419) a(420)
%e A359005 8 | a(47) a(48) a(103) a(104) a(151) a(152) a(440) a(441) a(442)
%e A359005 9 | a(58) a(59) a(244) a(245) a(246) a(247) a(248) a(249) a(250)
%e A359005 10 | a(34) a(35) a(36) a(37) a(38) a(39) a(958) a(959) a(960)
%e A359005 11 | a(45) a(46) a(212) a(213) a(520) a(521) a(522) a(1455) a(1456)
%e A359005 12 | a(56) a(57) a(242) a(243) a(290) a(291) a(785) a(786) a(787)
%e A359005 13 | a(21) a(22) a(299) a(300) a(301) a(302) a(647) a(648) a(3437)
%e A359005 .. |
%e A359005 .
%e A359005 The later illustration helps visualize how {a(n)} works: going from a(n) to a(n+1) results in either
%e A359005 - jumping forwards, thus moving one step to the right in the row of the table, or
%e A359005 - jumping backwards, thus changing row.
%e A359005 This procedure (demonstrably) never creates a gap in the rows of the table.
%o A359005 (Python)
%o A359005 def a(n: int) -> int:
%o A359005 if n < 0: raise Exception("n must be a nonnegative integer")
%o A359005 if n == 0: return 0
%o A359005 visited = {0}
%o A359005 slab = 1
%o A359005 for i in range(1, n):
%o A359005 label = 1 + (slab >> 1)
%o A359005 if not slab - label in visited:
%o A359005 slab -= label # moving backwards
%o A359005 else:
%o A359005 slab += label # moving forwards
%o A359005 if slab in visited: raise Exception(f"blocked at slab {slab}")
%o A359005 visited.add(slab)
%o A359005 return slab
%Y A359005 A particular valid path on the Jane Street infinite sidewalk A358838.
%Y A359005 Possibly the reverse mapping of sequence A359008.
%Y A359005 Cf. A006999.
%K A359005 nonn
%O A359005 0,3
%A A359005 _Frederic Ruget_, Dec 10 2022