A097417

[Benoit Cloitre]

Hi,

This non linear recursive Quet-sequence has interesting properties to 
me :

%S A097417 
1,1,2,5,13,34,90,236,621,1629,4274,11193,29337,76818,201173,526730,
%T A097417 
1379178,3610804,9453695,24750281,64798235,169644626,444138288,
%U A097417 
1162770238,3044180080,7969770106,20865148382,54625676431,143011928942
%N A097417 a(1)=1; a(n+1) = sum{k=1 to n} a(k) a(floor(n/k)).
%C A097417 Ratio a(n+1)/a(n) seems to tend to 1+Golden Ratio = 
2.61803398... = 1 + A001622 - Mark Hudson

since the sequence satisfy the "partial linear recursion" :

a(prime(n)+1) = 3*a(prime(n))- a(prime(n)-1)

(if a(n+1)=3*a(n)-a(n-1) for n>4 n is prime.)

this explains why we get : a(n+1)/a(n)-->1+Phi but lim n-->infty 
a(n)/(1+Phi)^n which exists should not have simple closed form.

Similarly :

a(1)=1; a(n+1) = sum{k=1 to n} a(k) a(ceiling(n/k))

leads to the partial linear recursion (if I'm not wrong) :

a(prime(n)+2) = 3*a(prime(n)+1)- a(prime(n)) and seems not to be in the 
OEIS.

Benoit
-------------- next part --------------
A non-text attachment was scrubbed...
Name: not available
Type: text/enriched
Size: 1067 bytes
Desc: not available
URL: <http://list.seqfan.eu/pipermail/seqfan/attachments/20040829/00bf024a/attachment.bin>