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A084978
Number of ways to represent n as a+b*(c+d*(e+f*(...x+y*(z)...))) in positive integers.
5
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
FORMULA
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
MATHEMATICA
a[1] = 1; a[n_] := a[n] = a[n-1] + DivisorSum[n-1, a[#] &];
Array[a, 50] (* Paolo Xausa, Aug 24 2024 *)
PROG
(PARI) first(upto) = {my(a=vector(upto, i, 1)); for(n=1, upto-1, a[n+1]=a[n]+sumdiv(n, d, a[d])); a} \\ Jason Yuen, Aug 24 2024
CROSSREFS
Sequence in context: A134527 A124379 A302830 * A118036 A291552 A208739
KEYWORD
nonn,easy
AUTHOR
David W. Wilson, Jun 16 2003
STATUS
approved