A281131 -id:A281131 - cp's OEIS frontend

A281130 a(n+1) = a(n-a(n)) if a(n-1) < a(n), otherwise a(n+1) = 2*a(n); a(1) = a(2) = 1.

1, 1, 2, 1, 2, 2, 4, 2, 4, 2, 4, 4, 8, 2, 4, 4, 8, 4, 8, 4, 8, 8, 16, 4, 8, 8, 16, 4, 8, 8, 16, 4, 8, 8, 16, 8, 16, 8, 16, 16, 32, 4, 8, 16, 4, 8, 8, 16, 4, 8, 4, 8, 16, 16, 32, 16, 32, 8, 16, 8, 16, 4, 8, 32, 4, 8, 8, 16, 8, 16, 16, 32, 16, 32, 4, 8, 16, 16, 32, 8

Offset: 1

Rok Cestnik, Jan 15 2017

nonn

The sequence is well defined, namely a(n) < n (with the exception of a(1)). Proof: Suppose a(s) = s+m, with m >= 0, is the first occurrence of a(n) >= n. It follows that a(s-1) = a(s-2) = (s+m)/2 and from there it follows that a(s-2-(s+m)/2) = (s+m)/2, but s-2-(s+m)/2 < (s+m)/2 which is a contradiction to the first statement. - Rok Cestnik, Aug 29 2017
From Robert G. Wilson v, Jan 16 2017: (Start)
a(n) = 2^k, k >= 0.
a(n)=1 for n: 1, 2, 4;
a(n)=2 for n: 3, 5, 6, 8, 10, 14;
a(n)=4 for n: 7, 9, 11, 12, 15, 16, 18, 20, 24, 28, 32, 42, 45, 49, 51, 62, 65, 75, 82, 84, 101, 108, 110, 118, 127, 175, 240;
First occurrence of 2^k (A281131): 1, 3, 7, 13, 23, 41, 98, 223, 437, 699, 1213, 2624, 4674, 11163, 21300, 40858, 73977, etc.;
Last occurrence of 2^k: 4, 14, 240, 1314, 10565, 35893, 62417, 638149, 2030926, etc.;
Number of occurrences of 2^k: 3, 6, 27, 77, 167, 330, 706, 1756, 3811, etc.
(End)

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

Rok Cestnik, Table of n, a(n) for n = 1..10000
Rok Cestnik, Self-referencing visualization

Cf. A281131, A291598.

#include
#include
int main(void){
   int N = 1000;
   int *a = (int*)malloc((N+1)*sizeof(int));
   printf("1 1\n2 1\n");
   a[1] = 1;
   a[2] = 1;
   for(int i = 2; i < N; ++i){
      if(a[i-1] < a[i]) a[i+1] = a[i-a[i]];
      else a[i+1] = 2*a[i];
      printf("%d %d\n", i+1, a[i+1]);
   }
   return 0;
}

a[n_] := a[n] = If[a[n -2] < a[n -1], a[n -1 -a[n -1]], 2 a[n -1]]; a[1] = a[2] = 1; Array[a, 100]

A291598 a(n) = log_2(A281130(n)).

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 2, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 3, 4, 4, 5, 2, 3, 4, 2, 3, 3, 4, 2, 3, 2, 3, 4, 4, 5, 4, 5, 3, 4, 3, 4, 2, 3, 5, 2, 3, 3, 4, 3, 4, 4, 5, 4, 5, 2, 3, 4, 4, 5, 3, 4, 2, 3, 2, 3, 4, 4, 5, 4, 5, 3, 4, 3, 4, 4, 5, 5, 6

Offset: 1

Rok Cestnik, Aug 27 2017

nonn

Rok Cestnik, Table of n, a(n) for n = 1..10000
Rok Cestnik, Self-referencing visualization
Rok Cestnik, Digit probability distributions

Cf. A281130, A281131.

#include
#include
#include
int main(void){
   int N = 100;
   int *a = (int*)malloc((N+1)*sizeof(int));
   printf("1 0\n2 0\n");
   a[1] = 0;
   a[2] = 0;
   for(int i = 2; i < N; ++i){
      if(a[i-1] < a[i]) a[i+1] = a[i-(int)(pow(2,a[i]))];
      else a[i+1] = a[i]+1;
      printf("%d %d\n", i+1, a[i+1]);
   }
   return 0;
}

a[n_] := a[n] = If[a[n - 2] < a[n - 1], a[n - 1 - a[n - 1]], 2 a[n - 1]]; a[1] = a[2] = 1; Array[Log2@ a@ # &, 105] (* Michael De Vlieger, Aug 29 2017 *)

cp's OEIS Frontend

A281130 a(n+1) = a(n-a(n)) if a(n-1) < a(n), otherwise a(n+1) = 2*a(n); a(1) = a(2) = 1.

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