A353651 Unique monotonic sequence of positive integers satisfying a(a(n)) = k*(n-1) + 3, where k = 4.

2, 3, 7, 8, 9, 10, 11, 15, 19, 23, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 47, 51, 55, 59, 63, 67, 71, 75, 79, 83, 87, 91, 95, 99, 103, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125

Offset: 1

Author

Yifan Xie, May 02 2022

Keywords

• nonn
• easy

Comments

Numbers m such that the base-4 representation of (3*m-1) starts with 11 or 12 or 13 or ends with 0.

First differences give a run of 4^i 1's followed by a run of 4^i 4's, for i = 0, 1, 2, ...

Examples

			a(6) = 10 because (2*4^1 + 1)/3 < 6 <= (5*4^1 + 1)/3, hence a(6) = 6 + 4^1 = 10;
a(9) = 19 because (5*4^1 + 1)/3 < 9 <= (8*4^1 + 1)/3, hence a(9) = 4*(9 - 4^1) - 1 = 19.
		

Links

• Yifan Xie, Table of n, a(n) for n = 1..10000
• Index entries for sequences of the a(a(n)) = 2n family

Crossrefs

For other values of k: A080637 (k=2), A003605 (k=3), this sequence (k=4), A353652 (k=5), A353653 (k=6).

Programs

Maple

isA353651 := proc(n)
    if modp(n,4) = 3 then
        true;
    else
        b4 := convert(3*n-1,base,4) ;
        if op(-1,b4) = 1 and op(-2,b4) <> 0  then
            true ;
        else
            false;
        end if;
    end if;
end proc:
for n from 2 to 122 do
    if isA353651(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Jul 05 2022

PARI

a(n) = my(n3=3*n, s=logint(n3>>1, 4)<<1); if(n3>>s < 5, n + 1<Kevin Ryde, Apr 15 2022
(C++)
/* program used to generate the b-file */
#include
using namespace std;
int main(){
    int cnt1=1, flag=0, cnt2=1, a=2;
    for(int n=1; n<=10000; n++) {
        cout<
				

Formula

For n in the range (2*4^i + 1)/3 < n <= (5*4^i + 1)/3, for i >= 0:

a(n) = n + 4^i.

a(n) = 1 + a(n-1).

Otherwise, for n in the range (5*4^i + 1)/3 < n <= (8*4^i + 1)/3, for i >= 0:

a(n) = 4*(n - 4^i) - 1.

a(n) = 4 + a(n-1).