A084978 Number of ways to represent n as a+b*(c+d*(e+f*(...x+y*(z)...))) in positive integers.

1, 2, 5, 11, 25, 51, 110, 221, 456, 918, 1864, 3729, 7528, 15057, 30227, 60485, 121205, 242411, 485337, 970675, 1942307, 3884730, 7771327, 15542655, 31089330, 62178686, 124364903, 248730268, 497475717, 994951435, 1989934099, 3979868199

Offset

1,2

Links

Robert Israel, Table of n, a(n) for n = 1..3305

Formula

a(n+1) = Sum(k=1..n; Sum(d|k; a(d)).

a(n+1) = 1 + Sum(k=1..n; Sum(d|k; a(d)) or a(n+1) = a(n) + Sum(d|n; a(d)). - Vladeta Jovovic, Aug 08 2004

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

a(n) ~ c * 2^n, where c = 0.9266495520163897954172886595779644507444342903568129920731434938761... - Vaclav Kotesovec, Dec 26 2023

Example

a(3) = 5: 3 = 2+1(1) = 1+2(1) = 1+1(2) = 1+1(1+1(1)).

Maple

f:= proc(n) option remember; local d;
   procname(n-1) + add(procname(d), d = numtheory:-divisors(n-1))
end proc:
f(1):=1:
map(f, [$1..40]); # Robert Israel, Dec 25 2023

Crossrefs

Sequence in context: A134527 A124379 A302830 * A118036 A291552 A208739

Adjacent sequences: A084975 A084976 A084977 * A084979 A084980 A084981

Keyword

nonn

Author

David W. Wilson, Jun 16 2003

Status

approved