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

%I A084978 #22 Aug 25 2024 18:46:03
%S A084978 1,2,5,11,25,51,110,221,456,918,1864,3729,7528,15057,30227,60485,
%T A084978 121205,242411,485337,970675,1942307,3884730,7771327,15542655,
%U A084978 31089330,62178686,124364903,248730268,497475717,994951435,1989934099,3979868199
%N A084978 Number of ways to represent n as a+b*(c+d*(e+f*(...x+y*(z)...))) in positive integers.
%H A084978 Robert Israel, <a href="/A084978/b084978.txt">Table of n, a(n) for n = 1..3305</a>
%F A084978 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
%F A084978 G.f. A(x) satisfies: A(x) = (x/(1 - x)) * (1 + Sum_{k>=1} A(x^k)). - _Ilya Gutkovskiy_, Feb 25 2020
%F A084978 a(n) ~ c * 2^n, where c = 0.9266495520163897954172886595779644507444342903568129920731434938761... - _Vaclav Kotesovec_, Dec 26 2023
%e A084978 a(3) = 5: 3 = 2+1(1) = 1+2(1) = 1+1(2) = 1+1(1+1(1)).
%p A084978 f:= proc(n) option remember; local d;
%p A084978    procname(n-1) + add(procname(d), d = numtheory:-divisors(n-1))
%p A084978 end proc:
%p A084978 f(1):=1:
%p A084978 map(f, [$1..40]); # _Robert Israel_, Dec 25 2023
%t A084978 a[1] = 1; a[n_] := a[n] = a[n-1] + DivisorSum[n-1, a[#] &];
%t A084978 Array[a, 50] (* _Paolo Xausa_, Aug 24 2024 *)
%o A084978 (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
%K A084978 nonn,easy
%O A084978 1,2
%A A084978 _David W. Wilson_, Jun 16 2003