Michael Thomas De Vlieger, St. Louis, Missouri, 2022 1122.
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The Van Eck sequence V = A181391 is a sort of self-cataloging sequence defined as follows.
The sequence begins with V(1) = 0. Suppose V(k) = m. We set c(m) = k only upon index k+1, so as to maintain record of the penultimate occasion of m. Therefore, if V(n−1) = m, then V(n) = n − c(m) − 1. This sequence is easily generated by Code 1.
The sequence begins as follows:
0, 0, 1, 0, 2, 0, 2, 2, 1, 6, 0, 5, 0, 2, 6, 5, 4, 0, 5, 3, 0, 3, 2, 9, 0, 4, 9, 3, 6, 14, 0, 6, 3, 5, 15, 0, 5, 3, 5, 2, 17, 0, 6, 11, 0, 3, 8, 0, 3, 3, 1, 42, 0, 5, 15, 20, 0, 4, 32, 0, 3, 11, 18, 0, 4, 7, 0, 3, 7, 3, 2, 31, 0, 6, 31, 3, 6, 3, 2, 8, 33, 0, 9, 56, 0, 3, 8, 7, 19, 0, 5, 37, 0, 3, 8, 8, 1, 46, 0, 6, 23, 0, 3, 9, 21, 0, 4, 42, 56, 25, 0, 5, 21, 8, 18, 52, 0, 6, 18, 4, ...
We can write a pair of sequences A and B such that V(B(ℓ)) = A(ℓ), the first ℓ-bit number in V. These sequences are akin to a pair of sequences of G. L. Honaker, Jr. relating to the smallest number of k decimal digits in V. The sequences are generated by Codes 2 and 3, respectively.
These sequences are considered simply to try to understand how V(n) progresses as n increases, with reference to binary (or decimal in the case of the Honaker sequences).
Table 1 lists A(n) and B(n) for n = 1…24. We show the binary expansion of these terms merely to attempt to discern any pattern that might emerge in the bits of the numbers that emerge.
n A(n) A(n)_2 n B(n) B(n)_2 n
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1 0 . 1 1 1 1
2 2 1. 2 5 1.1 2
3 6 11. 3 10 1.1. 3
4 9 1..1 4 24 11... 4
5 17 1...1 5 41 1.1..1 5
6 42 1.1.1. 6 52 11.1.. 6
7 92 1.111.. 7 152 1..11... 7
8 131 1.....11 8 162 1.1...1. 8
9 307 1..11..11 9 364 1.11.11.. 9
10 650 1.1...1.1. 10 726 1.11.1.11. 10
11 1024 1.......... 11 1150 1...111111. 11
12 2238 1...1.11111. 12 2451 1..11..1..11 12
13 4164 1.....1...1.. 13 4626 1..1....1..1. 13
14 8226 1.......1...1. 14 9847 1..11..111.111 14
15 17384 1....11111.1... 15 18131 1...11.11.1..11 15
16 33197 1......11.1.11.1 16 36016 1...11..1.11.... 16
17 67167 1.....11..1.11111 17 71709 1...11......111.1 17
18 133549 1.....1..11.1.11.1 18 143848 1...11...1111.1... 18
19 269119 1.....11.11..111111 19 276769 1....111..1..1....1 19
20 525974 1........11.1..1.11. 20 551730 1....11.1.11..11..1. 20
21 1055175 1.......11..111...111 21 1086371 1....1..1..111.1...11 21
22 2111641 1.......111...1..11..1 22 2158296 1.....111.111.11.11... 22
23 4213053 1.......1..1..1..1111.1 23 4297353 1.....11..1..1.1...1..1 23
24 8444257 1.......11.11..1.11....1 24 8607525 1.....11.1.1.111..1..1.1 24
Beyond a potential to perhaps surmise the second most significant 1-bit in the index B(n), there doesn't seem to be any pattern at all.
This concludes our examination.
a181391 =
Block[{nn, a, c, j, m}, nn = 2^20; a[_] = 0; c[_] = -1; c[0] = 2; m = 1;
Monitor[Do[Set[{j, a[n], c[m], m}, {c[m], m, n, 0}]; If[j > 0, m = n - j], {n, 3, nn}], n]; Array[a, nn]]
nn = 2^20; q[_] = False; q[0] = True; a[_] = 0; c[_] = -1; c[0] = 2; m = 1; {0}~Join~Rest@ Reap[Do[j = c[m]; k = m; c[m] = n; m = 0; If[j > 0, m = n - j]; If[! q[#], Sow[k]; q[#] = True] & @ IntegerLength[k, 2], {n, 3, nn}] ][[-1, -1]]
nn = 2^20; q[_] = False; q[0] = True; a[_] = 0; c[_] = -1; c[0] = 2; m = 1; {1}~Join~Rest@ Reap[Do[j = c[m]; k = m; c[m] = n; m = 0; If[j > 0, m = n - j]; If[! q[#], Sow[n]; q[#] = True] & @ IntegerLength[k, 2], {n, 3, nn}] ][[-1, -1]]
A181391: The Van Eck sequence V.
A358168: The first n-digit decimal number to appear in V.
A358180: Index of the first n-digit decimal number to appear in V.
A358258: The first n-bit number to appear in V.
A358259: Index of the first n-bit number to appear in V.
2022 1122 1800.