There are two men named “Van Eck”. First, in 1985, showed the world how for $ 15 to intercept data from the monitor (Van Eck phreaking), the second, in 2010, came up with a cunning sequence (Van Eck”s sequence). Steeper ease of setting the sequences can only be its properties and mysteries.
So, the algorithm used to generate members of the sequence. Take the “starting number”, for example “0” is issued. The next member is, how many steps ago, I met this number in the previous sub-sequence. If never, then write zero. The next is how many steps ago, I met zero in the previous sub-sequence, that is, one step back. Recordable unit. The unit for the first time, write zero. Oops, zero met two steps back. Write two, and so on…
To point “0” of the first 97 members of the sequence:
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
Chart:
Even more chart:
Quite easily proved properties of the sequence that its maximum member increases all the time and that there are an infinite number of zeros. Or that there are no periods. (Some theorems and consequences here.)
Logarithmic chart:
Program in Python:
A181391 = [0] last_pos = {} for i in range(10**4): new_value = i - last_pos.get(A181391[i], i) A181391.append(new_value) last_pos[A181391[i]] = i # Ehsan Kia, 12 Jun 2019For a starting number “1” the first hundred is this:
1, 0, 0, 1, 3, 0, 3, 2, 0, 3, 3, 1, 8, 0, 5, 0, 2, 9, 0, 3, 9, 3, 2, 6, 0, 6, 2, 4, 0, 4, 2, 4, 2, 2, 1, 23, 0, 8, 25, 0, 3, 19, 0, 3, 3, 1, 11, 0, 5, 34, 0, 3, 7, 0, 3, 3, 1, 11, 11, 1, 3, 5, 13, 0, 10, 0, 2, 33, 0, 3, 9, 50, 0, 4, 42, 0, 3, 7, 25, 40, 0, 5, 20, 0, 3, 8, 48, 0, 4, 15
Chart:
For the start number “2” the first hundred is this:
2, 0, 0, 1, 0, 2, 5, 0, 3, 0, 2, 5, 5, 1, 10, 0, 6, 0, 2, 8, 0, 3, 13, 0, 3, 3, 1, 13, 5, 16, 0, 7, 0, 2, 15, 0, 3, 11, 0, 3, 3, 1, 15, 8, 24, 0, 7, 15, 5, 20, 0, 5, 3, 12, 0, 4, 0, 2, 24, 14, 0, 4, 6, 46, 0, 4, 4, 1, 26, 0, 5, 19, 0, 3, 21, 0, 3, 3, 1, 11, 42, 0, 6, 20, 34, 0, 4, 20, 4
Chart:
