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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.
3
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)
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
a(n+1) - a(n) = 4*A018902(n-3), n > 2.
Sequence in context: A302786 A322026 A363247 * A238326 A083480 A179547
KEYWORD
nonn,easy
AUTHOR
Paul D. Hanna, Aug 24 2003
STATUS
approved