A318281 -id:A318281 - cp's OEIS frontend

A318282 a(n) = (A318281(n) - 1)/3.

0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 2, 0, 2, 0, 2, 0, 2, 1, 0, 1, 0, 1, 1, 2, 2, 3, 0, 3, 1, 2, 1, 0, 1, 1, 2, 3, 3, 4, 2, 0, 2, 1, 4, 2, 3, 2, 2, 3, 4, 3, 0, 3, 1, 4, 2, 3, 2, 3, 3, 4, 2, 4, 4, 5, 3, 2, 3, 2, 2, 3, 4, 3, 4, 4, 5

Offset: 1

Rok Cestnik, Aug 23 2018

nonn

Rok Cestnik, Table of n, a(n) for n = 1..9999
Rok Cestnik, Visualization

Cf. A318281, A281130, A291598, A004001, A005185, A005229.

#include
#include
#include
int main(void){
  int N = 100; //number of terms
  int *a = (int*)malloc((N+1)*sizeof(int));
  printf("1 0\n2 0\n3 0\n4 0\n");
  a[1] = 0;
  a[2] = 0;
  a[3] = 0;
  a[4] = 0;
  for(int i = 4; i < N; ++i){
    if(a[i-1] != a[i]) a[i+1] = a[i-(3*a[i]+1)];
    else a[i+1] = a[i]+1;
    printf("%d %d\n", i+1, a[i+1]);
  }
  free(a);
  return 0;
}

A330631 a(n+1) = a(n-a(n))+1 if a(n) > a(n-1), otherwise a(n+1) = 2*a(n); a(1) = a(2) = 1.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 4, 3, 6, 3, 6, 5, 10, 3, 6, 7, 7, 14, 3, 6, 4, 8, 4, 8, 8, 16, 4, 8, 7, 14, 8, 16, 8, 16, 15, 30, 3, 6, 17, 9, 18, 5, 10, 9, 18, 5, 10, 4, 8, 19, 9, 18, 17, 34, 7, 14, 6, 12, 6, 12, 5, 10, 19, 10, 20, 19, 38, 8, 16, 18, 19, 19, 38, 16, 32

Offset: 1

Rok Cestnik, Dec 21 2019

nonn

a(n) < n (with exception a(1) = 1). Proof: Suppose a(s) = s+x, x >= 0, is the first occurrence of a(n) >= n. From here we branch into two possibilities: Possibility #1: a(s-2) < a(s-1), from which it follows that a(s) = a(s-1-a(s-1))+1 and therefore a(s-1-a(s-1)) = s-1+x is an earlier example of a(n) >= n. Possibility #2: a(s-2) >= a(s-1) and the terms can be expressed as a(s-1) = (s+x)/2 and a(s-2) = (s+x)/2+y with y >= 0. From this it follows that a(s-2-((s+x)/2+y))+1 = (s+x)/2, which when simplified reveals that a((s+x)/2-2-x-y) = (s+x)/2-1 is an earlier example of a(n) >= n. Both possibilities lead to a contradiction of the first statement, therefore we conclude that there is no occurrence of a(n) >= n (with exception a(1) = 1).

Some numbers never seem to appear in the sequence; the smallest of these are 328, 329, 331, 332, 333, 445, 667, 668, 669, ...

			a(3) = 2*a(2) = 2 because a(2) <= a(1).
a(4) = a(3-a(3))+1 = 2 because a(3) > a(2).

See A281130 for a similar sequence.

Cf. A004001, A005229, A055748, A318281.

Nest[Append[#, If[Less @@ Take[#, -2], #[[Length@ # - #[[-1]] ]] + 1, 2 #[[-1]] ]] &, {1, 1}, 73] (* Michael De Vlieger, Dec 23 2019 *)

a = [1,1]
for n in range(1, 1000):
    if(a[n] > a[n-1]):
        a.append(a[n-a[n]]+1)
    else:
        a.append(2*a[n])

cp's OEIS Frontend

A318282 a(n) = (A318281(n) - 1)/3.

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A330631 a(n+1) = a(n-a(n))+1 if a(n) > a(n-1), otherwise a(n+1) = 2*a(n); a(1) = a(2) = 1.

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