A014668 a(1) = 1, a(n) = Sum_{k=1..n-1} Sum_{d|k} a(d).

1, 1, 3, 7, 16, 33, 71, 143, 295, 594, 1206, 2413, 4871, 9743, 19559, 39138, 78428, 156857, 314047, 628095, 1256809, 2513693, 5028594, 10057189, 20116979, 40233975, 80472823, 160945945, 321901713, 643803427, 1287627061, 2575254123, 5150547536, 10301096282

Offset: 1

Author

Benoit Cloitre, Jun 24 2003

Keywords

• nonn

Comments

Equals eigensequence of triangle A010766 and starting (1, 3, 7, 16, 33, ...) = row sums of triangle A163313. - Gary W. Adamson, Jul 30 2009. Gary Adamson's comment may be restated as "This sequence shifts left by one place under the floor transform." - N. J. A. Sloane, Feb 05 2016

The Gould & Quaintance reference, published in 2007, says incorrectly that this sequence is not in the OEIS. - Olivier Gérard, Oct 20 2011

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000
• H. W. Gould and J. Quaintance, Floor and Roof function analog of the Bell Numbers, INTEGERS, 7 (2007), #A58.

Crossrefs

Cf. A010766, A163313. - Gary W. Adamson, Jul 30 2009

Programs

Maple

with(numtheory):
a:= proc(n) option remember;
      `if`(n=1, 1, add(add(a(d), d=divisors(k)), k=1..n-1))
    end:
seq(a(n), n=1..40);  # Alois P. Heinz, Oct 28 2011

Mathematica

a[1] = 1; a[n_] := a[n] = Sum[Sum[a[d], {d, Divisors[k]}], {k, 1, n-1}]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Apr 07 2015 *)

PARI

// an=vector(100); a(n)=if(n<0,0,an[n]); // an[1]=1; for(n=2,100,an[n]=sum(k=1,n-1,sumdiv(k,d,a(d))))

Formula

a(n) is asymptotic to c*2^n where c = 0.59960731361450033896934...

a(n+1) = Sum_{k=1..n} a(k)*floor(n/k). - Franklin T. Adams-Watters, Mar 21 2017

G.f. A(x) satisfies: A(x) = x * (1 + (1/(1 - x)) * Sum_{k>=1} A(x^k)). - Ilya Gutkovskiy, Feb 25 2020