A281130 - OEIS

%I #37 Sep 13 2017 03:33:09

%S 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,

%T 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,

%U 16,4,8,32,4,8,8,16,8,16,16,32,16,32,4,8,16,16,32,8

%N 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.

%C 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

%C From _Robert G. Wilson v_, Jan 16 2017: (Start)

%C a(n) = 2^k, k >= 0.

%C a(n)=1 for n: 1, 2, 4;

%C a(n)=2 for n: 3, 5, 6, 8, 10, 14;

%C 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;

%C First occurrence of 2^k (A281131): 1, 3, 7, 13, 23, 41, 98, 223, 437, 699, 1213, 2624, 4674, 11163, 21300, 40858, 73977, etc.;

%C Last occurrence of 2^k: 4, 14, 240, 1314, 10565, 35893, 62417, 638149, 2030926, etc.;

%C Number of occurrences of 2^k: 3, 6, 27, 77, 167, 330, 706, 1756, 3811, etc.

%C (End)

%H Rok Cestnik, <a href="/A281130/b281130.txt">Table of n, a(n) for n = 1..10000</a>

%H Rok Cestnik, <a href="/A281130/a281130.pdf">Self-referencing visualization</a>

%e a(3) = 2*a(2) = 2 because a(1) !< a(2).

%e a(4) = a(3-a(3)) = 1 because a(2) < a(3).

%t 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]

%o (C)

%o #include<stdio.h>

%o #include<stdlib.h>

%o int main(void){

%o    int N = 1000;

%o    int *a = (int*)malloc((N+1)*sizeof(int));

%o    printf("1 1\n2 1\n");

%o    a[1] = 1;

%o    a[2] = 1;

%o    for(int i = 2; i < N; ++i){

%o       if(a[i-1] < a[i]) a[i+1] = a[i-a[i]];

%o       else a[i+1] = 2*a[i];

%o       printf("%d %d\n", i+1, a[i+1]);

%o    }

%o    return 0;

%o }

%Y Cf. A281131, A291598.

%K nonn

%O 1,3

%A _Rok Cestnik_, Jan 15 2017