1,2

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.

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.

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

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)={prod(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.

a(n) is asymptotic to (2+sqrt2)^(1/(2+sqrt2))*n^(sqrt2-1); conjecture : a(n)=(2+sqrt2)^(1/(2+sqrt2))*n^(sqrt2-1) + O(1)

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

Cf. A000002, A001462, A088496.

Sequence in context: A130239 A091092 A083375 this_sequence A135034 A003059 A011752

Adjacent sequences: A088516 A088517 A088518 this_sequence A088520 A088521 A088522

nonn

Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 13 2003