1,3

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 (f.firoozbakht(AT)math.ui.ac.ir), Feb 05 2005

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 ->infty a(n)/(1+Phi)^n exists but should not have a simple closed form. - Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 29 2004

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) (from Mark Hudson)

a[1] = 1; a[n_] := a[n] = Sum[ a[k]*a[Floor[(n - 1)/k]], {k, n - 1}]; Table[ a[n], {n, 29}] (from Robert G. Wilson v Aug 21 2004)

(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

Cf. A097919, A038044, A078346.

Adjacent sequences: A097414 A097415 A097416 this_sequence A097418 A097419 A097420

Sequence in context: A099496 A114299 A112842 this_sequence A006801 A114173 A023425

easy,nonn

Leroy Quet (qq-quet(AT)mindspring.com), Aug 19 2004

More terms from Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Robert G. Wilson v (rgwv(AT)rgwv.com) and Mark Hudson (mrmarkhudson(AT)hotmail.com), Aug 21 2004