%I #20 Sep 23 2022 10:17:02

%S 1,2,3,4,1,5,2,6,1,3,7,2,1,4,8,3,1,2,5,9,1,4,2,3,1,6,10,2,1,5,3,4,1,2,

%T 7,11,1,3,2,6,1,4,5,2,1,3,8,12,1,2,4,3,1,7,2,5,1,6,3,2,1,4,9,13,1,2,3,

%U 5,1,4,2,8,1,3,6,2,1,7,4,3,1,2,5,10,1,14,2,3,1,4,6,2,1,5,3,9,1,2,4,7,1,3,2

%N a(n)=1 when n == 1 (mod 4), otherwise a(n) = a(n - ceiling(n/4)) + 1. Removing all the 1's results in the original sequence with every term incremented by 1.

%C Indices of records are given by A087192: a(A087192(n))=n, where A087192(n) = ceiling(A087192(n-1)*4/3).

%C From _Benoit Cloitre_, Mar 07 2009: (Start)

%C To construct the sequence:

%C Step 1: start from a sequence of 1's, leaving 3 undefined places between 1's, giving 1,(),(),(),1,(),(),(),1,(),(),(),1,(),(),(),1,(),(),(),1,...

%C Step 2: replace the first undefined place with a 2 and leave 3 undefined places between 2's, giving 1,2,(),(),1,(),2,(),1,(),(),2,1,(),(),(),1,2,(),(),1,...

%C Step 3: replace the first undefined place with a 3 and leave 3 undefined places between 3's, giving 1,2,3,(),1,(),2,(),1,3,(),2,1,(),(),3,1,2,(),(),1,...

%C Step 4: replace the first undefined place with a 4 and leave 3 undefined places between 4's, giving 1,2,3,4,1,(),2,(),1,3,(),2,1,4,(),3,1,2,(),(),1,...

%C Iterating the process indefinitely yields the sequence: 1,2,3,4,1,5,2,6,1,3,7,2,1,4,8,3,1,2,5,9,1,... (End)

%F a(n) = 4 + A244041(4*(n-1)) - A244041(4*n). - _Tom Edgar_ and _James Van Alstine_, Aug 05 2014

%F a(4*n) = a(3*n)+1.

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

%F a(4*n+2) = a(3*n+1)+1.

%F a(4*n+3) = a(3*n+2)+1. - _Robert Israel_, Aug 05 2014

%F a(n) < k*log(n) + 4 for n > 1 where k = 1/log(4/3) < 3.5. - _Charles R Greathouse IV_, Sep 22 2022

%p for n from 1 to 100 do

%p if n mod 4 = 1 then A[n]:= 1

%p else A[n]:= A[n - ceil(n/4)] + 1

%p fi

%p od:

%p seq(A[n],n=1..100); # _Robert Israel_, Aug 05 2014

%o (PARI) a(n)=my(s); while(n>4, if(n%4==1, return(s+1)); n=(n\4*3)+max(n%4 - 1,0); s++); s+n \\ _Charles R Greathouse IV_, Sep 22 2022

%Y Cf. A001511, A087088, A087192, A244041.

%Y a(n+1) - a(n) = 4*A018902(n-3), n > 2.

%K nonn,easy

%O 1,2

%A _Paul D. Hanna_, Aug 24 2003