

A088519


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


0



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, 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, 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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

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(n1)<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 kGolomb's sequence G(n,k). Let S(n,0)=G(n,k) and S(n,k) = Sum_{i=1..n} S(i,k1) the sequence G(n,k) such that G(1,k)=1, G(2,k)=2 and "length of nth run = S(n,k1)" is asymptotic to r(k)*n^s(k) where s(k)=(k+sqrt(k^2+4))/2 and r(k) = (Product_{i=0..k1} (1/s(k)i))^(s(k)/(1+s(k))). Golomb's sequence is obtained for k=1. Alternative description: the sequence of kth differences of the function "rank of the last occurrence of m in (G(n,k))n>=1" is (G(n,k))n>=1 itself.


LINKS

Table of n, a(n) for n=1..102.


FORMULA

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


EXAMPLE

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


CROSSREFS

Cf. A000002, A001462, A088496.
Sequence in context: A218461 A186189 A083375 * A135034 A003059 A325678
Adjacent sequences: A088516 A088517 A088518 * A088520 A088521 A088522


KEYWORD

nonn


AUTHOR

Benoit Cloitre, Nov 13 2003


STATUS

approved



