A248624 Hiccup sequence.

4, 5, 4, 6, 4, 5, 7, 11, 10, 5, 6, 8, 10, 14, 11, 21, 14, 6, 7, 9, 21, 13, 15, 23, 14, 5, 20, 33, 41, 17, 7, 8, 10, 20, 35, 14, 14, 23, 16, 33, 31, 51, 20, 33, 13, 32, 50, 45, 53, 24, 44, 8, 9, 11, 17, 33, 35, 12, 4, 35, 40, 17, 47, 48, 31, 14, 25, 20, 40, 27

Offset

2,1

Comments

The hiccup sequence #_n: Consider the sequences created by a_n = a_[n-1] + a_[n-2] in Z mod n, with a certain "carrying" rule, where when the numbers a_[n-2] and a_[n-1] are added together in Z and the result is greater than n, two numbers are added to the sequence, first 1, and then a_[n-2] + a_[n-1] mod n. These sequences all fall into cycles of a different length:

2: 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0... [4]

3: 0, 1, 1, 2, 1, 0, 1, 1, 2, 1, 0, 1, 1, 2, 1, 0, 1... [5]

4: 0, 1, 1, 2, 3, 1, 1, 2, 3, 1, 1, 2, 3, 1, 1, 2, 3... [4]

5: 0, 1, 1, 2, 3, 1, 0, 1, 1, 2, 3, 1, 0, 1, 1, 2, 3... [6]

6: 0, 1, 1, 2, 3, 5, 1, 2, 3, 5, 1, 2, 3, 5, 1, 2, 3... [4]

7: 0, 1, 1, 2, 3, 5, 1, 1, 2, 3, 5, 1, 1, 2, 3, 5, 1... [5]

8: 0, 1, 1, 2, 3, 5, 1, 0, 1, 1, 2, 3, 5, 1, 0, 1, 1... [7]

...and so on! It is easy to calculate and it seems to go on forever. But I do not have a proof.

Proof: This sequence continues forever. This is because the state of the sequence is totally determined by the previous two numbers, and it has only a finite number of states. Therefore, the same state of any carrying Fibonacci sequence must repeat after a finite period of time; in physics, this is Poincaré's recurrence theorem.

Proof: This sequence increases without bound. This is because in order to cycle a carrying Fibonacci sequence must wrap around at least once. Thus

#_n ≥ sup{k in N, Fib[k] < n}

Links

Lars Blomberg, Table of n, a(n) for n = 2..10000

Prog

(Lua)
for i = 2, 100 do
  local a = {i}
  for j = 1, 20 do
      local k
      if j <3 then
         k = j-1
      else
         k = a[#a] + a[#a-1]
      end
      if k >= i then
         a[#a+1] = 1
         k = k - i
      end
      a[#a+1] = k
  end
    print(table.concat(a, ", "))
end

Crossrefs

Cf. A248218, A001175.

Sequence in context: A223170 A303275 A198817 * A363195 A085428 A053025

Adjacent sequences: A248621 A248622 A248623 * A248625 A248626 A248627

Keyword

nonn

Author

Mason Bogue, Oct 10 2014

Extensions

Corrected a(9) and added a(13)-a(71) by Lars Blomberg, Oct 18 2014

Status

approved