A087165 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.

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, 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, 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

Offset

1,2

Comments

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

From Benoit Cloitre, Mar 07 2009: (Start)

To construct the sequence:

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

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,...

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,...

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,...

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)

Links

Table of n, a(n) for n=1..103.

Formula

a(n) = 4 + A244041(4*(n-1)) - A244041(4*n). - Tom Edgar and James Van Alstine, Aug 05 2014

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

a(4*n+1) = 1.

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

a(4*n+3) = a(3*n+2)+1. - Robert Israel, Aug 05 2014

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

Maple

for n from 1 to 100 do
  if n mod 4 = 1 then A[n]:= 1
  else A[n]:= A[n - ceil(n/4)] + 1
  fi
od:
seq(A[n], n=1..100); # Robert Israel, Aug 05 2014

Prog

(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

Crossrefs

Cf. A001511, A087088, A087192, A244041.

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

Sequence in context: A302786 A322026 A363247 * A238326 A083480 A179547

Adjacent sequences: A087162 A087163 A087164 * A087166 A087167 A087168

Keyword

nonn,easy

Author

Paul D. Hanna, Aug 24 2003

Status

approved