cp's OEIS Frontend

A298357 a(n) = a(n-1) + a(n-2) + a([n/2]) + a([n/3]) + ... + a([n/n]), where a(0) = 1, a(1) = 2, a(2) = 3.

%I A298357 #8 Mar 31 2021 17:24:39
%S A298357 1,2,3,9,19,37,74,131,238,410,710,1184,2014,3320,5516,9044,14888,
%T A298357 24262,39698,64510,105089,170545,277057,449027,728502,1179967,1912216,
%U A298357 3096110,5014519,8116824,13141430,21268343,34425826,55710704,90162442,145899135,236104060
%N A298357 a(n) = a(n-1) + a(n-2) + a([n/2]) + a([n/3]) + ... + a([n/n]), where a(0) = 1, a(1) = 2, a(2) = 3.
%C A298357 a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio (A001622), so that (a(n)) has the growth rate of the Fibonacci numbers (A000045). See A298338 for a guide to related sequences.
%H A298357 Clark Kimberling, <a href="/A298357/b298357.txt">Table of n, a(n) for n = 0..1000</a>
%t A298357 a[0] = 1; a[1] = 2; a[2] = 3;
%t A298357 a[n_] := a[n] = a[n - 1] + a[n - 2] + Sum[a[Floor[n/k]], {k, 2, n}];
%t A298357 Table[a[n], {n, 0, 30}]  (* A298357 *)
%o A298357 (Python)
%o A298357 from functools import lru_cache
%o A298357 @lru_cache(maxsize=None)
%o A298357 def A298357(n):
%o A298357     if n <= 2:
%o A298357         return n+1
%o A298357     c, j = A298357(n-1)+A298357(n-2), 2
%o A298357     k1 = n//j
%o A298357     while k1 > 1:
%o A298357         j2 = n//k1 + 1
%o A298357         c += (j2-j)*A298357(k1)
%o A298357         j, k1 = j2, n//j2
%o A298357     return c+2*(n-j+1) # _Chai Wah Wu_, Mar 31 2021
%Y A298357 Cf. A001622, A000045, A298338.
%K A298357 nonn,easy
%O A298357 0,2
%A A298357 _Clark Kimberling_, Feb 10 2018