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

1,3

Comments

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)

Links

Rok Cestnik, Table of n, a(n) for n = 1..10000

Rok Cestnik, Self-referencing visualization

Example

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

Mathematica

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]

Prog

(C)
#include<stdio.h>
#include<stdlib.h>
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;
}

Crossrefs

Cf. A281131, A291598.

Sequence in context: A139318 A205779 A205551 * A054541 A277891 A102722

Adjacent sequences: A281127 A281128 A281129 * A281131 A281132 A281133

Keyword

nonn

Author

Rok Cestnik, Jan 15 2017

Status

approved