login
2-Golomb's sequence G(n,2): earliest positive increasing sequence starting with (1,2) and satisfying "length of n-th run = n-th partial sum".
0

%I #11 Sep 01 2019 02:31:20

%S 1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,

%T 6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,

%U 8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10

%N 2-Golomb's sequence G(n,2): earliest positive increasing sequence starting with (1,2) and satisfying "length of n-th run = n-th partial sum".

%C Alternative definition: if A(n)=a(1)+...+a(n), (a(n))n>=1 satisfies a(1)=1, a(2)=2 and for A(1)+...+A(n-1)<m<=A(1)+...+A(n) a(m)=n.

%C Alternative description: unique positive increasing sequence (a(n))n>=1 starting with (1,2) and such that the sequence of second differences of the function "rank of the last occurrence of m in (a(n))n>=1" is (a(n))n>=1 itself.

%C Sequence has same kind of asymptotic behavior as Golomb's sequence.

%C Sequence is case k=2 of the following possible generalization of Golomb's sequence, say k-Golomb's sequence G(n,k). Let S(n,0)=G(n,k) and S(n,k) = Sum_{i=1..n} S(i,k-1) the sequence G(n,k) such that G(1,k)=1, G(2,k)=2 and "length of n-th run = S(n,k-1)" is asymptotic to r(k)*n^s(k) where s(k)=(-k+sqrt(k^2+4))/2 and r(k) = (Product_{i=0..k-1} (1/s(k)-i))^(s(k)/(1+s(k))). Golomb's sequence is obtained for k=1. Alternative description: the sequence of k-th differences of the function "rank of the last occurrence of m in (G(n,k))n>=1" is (G(n,k))n>=1 itself.

%F a(n) is asymptotic to (2+sqrt(2))^(1/(2+sqrt(2)))*n^(sqrt(2)-1);

%F conjecture: a(n) = (2+sqrt(2))^(1/(2+sqrt(2)))*n^(sqrt(2)-1) + O(1).

%e a(1) + a(2) + a(3) = 1+2+2 = 5, hence the third run has length 5 and consists of 5 3's.

%Y Cf. A000002, A001462, A088496.

%K nonn

%O 1,2

%A _Benoit Cloitre_, Nov 13 2003