A097919 -id:A097919 - cp's OEIS frontend

A097417 a(1)=1; a(n+1) = Sum_{k=1..n} a(k) a(floor(n/k)).

1, 1, 2, 5, 13, 34, 90, 236, 621, 1629, 4274, 11193, 29337, 76818, 201173, 526730, 1379178, 3610804, 9453695, 24750281, 64798235, 169644626, 444138288, 1162770238, 3044180080, 7969770106, 20865148382, 54625676431, 143011928942

Offset: 1

Leroy Quet, Aug 19 2004

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

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A038044, A078346, A097919.

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

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 *)

{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

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

More terms from Klaus Brockhaus, Robert G. Wilson v and Mark Hudson (mrmarkhudson(AT)hotmail.com), Aug 21 2004

A332846 a(1) = 1; a(n+1) = Sum_{k=1..n} a(k) * ceiling(n/k).

Original entry on oeis.org

1, 1, 3, 8, 20, 50, 121, 297, 716, 1739, 4198, 10157, 24513, 59246, 143006, 345381, 833792, 2013272, 4860337, 11734717, 28329772, 68396030, 165121957, 398644144, 962410246, 2323475153, 5609360573, 13542220814, 32693802921, 78929886033, 190553574988, 460037180829, 1110627936647

Offset: 1

Ilya Gutkovskiy, Feb 26 2020

nonn

Cf. A006590, A014668, A097919, A332490.

a[1] = 1; a[n_] := a[n] = Sum[a[k] Ceiling[(n - 1)/k], {k, 1, n - 1}]; Table[a[n], {n, 1, 33}]
a[1] = 1; a[n_] := a[n] = a[n - 1] + Sum[a[k] + Sum[a[d], {d, Divisors[k]}], {k, 1, n - 2}]; Table[a[n], {n, 1, 33}]
terms = 33; A[] = 0; Do[A[x] = x (1 + (1/(1 - x)) (A[x] + x Sum[A[x^k], {k, 1, terms}])) + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x] // Rest

G.f. A(x) satisfies: A(x) = x * (1 + (1/(1 - x)) * (A(x) + x * Sum_{k>=1} A(x^k))).
a(1) = 1; a(n) = a(n-1) + Sum_{k=1..n-2} (a(k) + Sum_{d|k} a(d)).
a(n) ~ c * (1 + sqrt(2))^n, where c = 0.2594006517235012546870541901936538347053403598092060748627156661727... - Vaclav Kotesovec, Mar 10 2020

cp's OEIS Frontend

A097417 a(1)=1; a(n+1) = Sum_{k=1..n} a(k) a(floor(n/k)).

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