A152009 (L)-sieve transform of {1,4,7,10,...,3n-2,...} (A016777)

1, 3, 6, 10, 16, 25, 39, 60, 91, 138, 208, 313, 471, 708, 1063, 1596, 2395, 3594, 5392, 8089, 12135, 18204, 27307, 40962, 61444, 92167

Offset

1,2

Comments

The (L)-sieve transform of the sequence {a(n)} of positive integers is defined as follows: Denote the sequence of natural numbers by N. Remove the first term of N, which we denote by s(1) and then from the resulting sequence delete all terms whose index is a term of {a(n)}, to obtain the sequence N'.

Then remove the first term of N', denoted by s(2) and then from the resulting sequence delete all terms whose index is a term of {a(n)}, to obtain N''. Repeat this process indefinitely to obtain the transform LST({a(n)}) = {s(1), s(2),...}, the sequence of initial terms removed at each stage.

The (L)-sieve transform is quite different from the transform introduced by N. J. A. Sloane in A099361 and used by T. D. Noe in A100424 - A100426 and seems to lead to more interesting results and relationships among sequences. An interesting property of the (L)-sieve transform is that the (L)-sieve transform of the sequence {1,3,6,10,...,n(n+1)/2,...} of triangular numbers is again the triangular numbers. Another (conjectured) connection with the triangular numbers is given in the following.

Conjecture. Let x(0) be a random sequence of positive integers and, for n>0, let x(n)=S[x(n-1)], where S is the (L)-sieve transform. Then the limit of {x(n)} as n goes to infinity is the sequence of triangular numbers {1,3,6,10,...,n(n+1)/2,...}.

Illustration of the conjecture:

x(0)={3,8,12,14,18,22,25,31,34,39,42,45,...} (A random initial sequence.)

x(1)={1,2,3,5,7,10,14,20,28,38,51,69,...}

x(2)={1,5,12,20,30,41,53,65,78,91,105,119,...}

x(3)={1,3,5,8,11,15,19,24,29,35,41,48,...}

x(4)={1,3,7,13,21,31,43,56,71,88,107,127,...}

x(5)={1,3,6,10,15,20,26,33,40,48,56,65,...}

x(6)={1,3,6,10,15,22,30,39,50,62,75,90,...}

x(7)={1,3,6,10,15,21,28,36,45,55,66,78,...} ...

t={1,3,6,10,15,21,28,36,45,55,66,78,...} (Triangular numbers)

Links

Table of n, a(n) for n=1..26.

Formula

It appears that {a(n)} is given by a(n)=floor[(3*a(n-1)+3)/2], with a(1)=1.

Prog

(Maxima)
a[1]:1$
a[n]:=floor((3*a[n-1]+3)/2)$
A152009(n):=a[n];
makelist(A152009(n), n, 1, 30); /* Martin Ettl, Oct 31 2012 */

Crossrefs

Sequence in context: A025004 A145131 A265072 * A255875 A114324 A265073

Adjacent sequences: A152006 A152007 A152008 * A152010 A152011 A152012

Keyword

nonn

Author

John W. Layman, Nov 19 2008

Status

approved