OFFSET
1,3
COMMENTS
4 is the only composite number n such that a(n+1) = 3a(n) - a(n-1) and if n is a composite number greater than 4 then a(n+1) > 3a(n) - a(n-1). - Farideh Firoozbakht, Feb 05 2005
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
FORMULA
Ratio a(n+1)/a(n) seems to tend to 1 + Golden Ratio = 2.61803398... = 1 + A001622. - Mark Hudson (mrmarkhudson(AT)hotmail.com), Aug 23 2004
Satisfies the "partial linear recursion": a(prime(n)+1) = 3*a(prime(n)) - a(prime(n)-1). This explains why we get a(n+1)/a(n) -> 1 + phi. Also, lim_{n->oo} a(n)/(1 + phi)^n exists but should not have a simple closed form. - Benoit Cloitre, Aug 29 2004
Limit_{n->oo} a(n)/(1 + phi)^n = 0.108165624886204570982244311730754895284041534583990405146651275318889227986... - Vaclav Kotesovec, May 28 2021
MAPLE
a[1]:=1: for n from 1 to 50 do: a[n+1]:=sum(a[k]*a[floor(n/k)], k=1..n): od: seq(a[i], i=1..51) # Mark Hudson, Aug 21 2004
MATHEMATICA
a[1] = 1; a[n_] := a[n] = Sum[ a[k]*a[Floor[(n - 1)/k]], {k, n - 1}]; Table[ a[n], {n, 29}] (* Robert G. Wilson v, Aug 21 2004 *)
PROG
(PARI) {m=29; a=vector(m); print1(a[1]=1, ", "); for(n=1, m-1, print1(a[n+1]=sum(k=1, n, a[k]*a[floor(n/k)]), ", "))} \\ Klaus Brockhaus, Aug 21 2004
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Leroy Quet, Aug 19 2004
EXTENSIONS
More terms from Klaus Brockhaus, Robert G. Wilson v and Mark Hudson (mrmarkhudson(AT)hotmail.com), Aug 21 2004
STATUS
approved