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Unique monotonic sequence of positive integers satisfying a(a(n)) = k*(n-1) + 3, where k = 4.
5

%I #47 Jul 16 2022 11:56:53

%S 2,3,7,8,9,10,11,15,19,23,27,28,29,30,31,32,33,34,35,36,37,38,39,40,

%T 41,42,43,47,51,55,59,63,67,71,75,79,83,87,91,95,99,103,107,108,109,

%U 110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125

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

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

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

%H Yifan Xie, <a href="/A353651/b353651.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Aa#aan">Index entries for sequences of the a(a(n)) = 2n family</a>

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

%F a(n) = n + 4^i.

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

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

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

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

%e a(6) = 10 because (2*4^1 + 1)/3 < 6 <= (5*4^1 + 1)/3, hence a(6) = 6 + 4^1 = 10;

%e a(9) = 19 because (5*4^1 + 1)/3 < 9 <= (8*4^1 + 1)/3, hence a(9) = 4*(9 - 4^1) - 1 = 19.

%p isA353651 := proc(n)

%p if modp(n,4) = 3 then

%p true;

%p else

%p b4 := convert(3*n-1,base,4) ;

%p if op(-1,b4) = 1 and op(-2,b4) <> 0 then

%p true ;

%p else

%p false;

%p end if;

%p end if;

%p end proc:

%p for n from 2 to 122 do

%p if isA353651(n) then

%p printf("%d,",n) ;

%p end if;

%p end do: # _R. J. Mathar_, Jul 05 2022

%o (PARI) a(n) = my(n3=3*n, s=logint(n3>>1, 4)<<1); if(n3>>s < 5, n + 1<<s, 4*(n - 1<<s) - 1); \\ _Kevin Ryde_, Apr 15 2022

%o (C++)

%o /* program used to generate the b-file */

%o #include<iostream>

%o using namespace std;

%o int main(){

%o int cnt1=1, flag=0, cnt2=1, a=2;

%o for(int n=1; n<=10000; n++) {

%o cout<<n<<" "<<a<<endl;

%o if(cnt2==cnt1) {

%o flag=1-flag, cnt1=1;

%o if(flag) a+=1;

%o else {

%o a+=4;

%o cnt2*=4;

%o }

%o }

%o else {

%o cnt1++;

%o a+=(flag?4:1);

%o }

%o }

%o return 0;

%o }

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

%K nonn,easy

%O 1,1

%A _Yifan Xie_, May 02 2022