cp's OEIS Frontend

A298370 a(n) = a(n-1) + a(n-2) + 2 a(floor(n/2)) + 3 a(floor(n/3)) + ... + n a(floor(n/n)), where a(0) = 1, a(1) = 2, a(2) = 3.

%I A298370 #10 Mar 31 2021 19:10:16
%S A298370 1,2,3,15,38,83,190,356,695,1254,2267,3861,6829,11417,19340,32076,
%T A298370 53545,87784,145048,236589,387765,631106,1028866,1670013,2716595,
%U A298370 4404599,7148426,11582096,18776334,30404300,49256015,79735758,129111774,208972513,338277831
%N A298370 a(n) = a(n-1) + a(n-2) + 2 a(floor(n/2)) + 3 a(floor(n/3)) + ... +  n a(floor(n/n)), where a(0) = 1, a(1) = 2, a(2) = 3.
%C A298370 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 A298370 Clark Kimberling, <a href="/A298370/b298370.txt">Table of n, a(n) for n = 0..1000</a>
%t A298370 a[0] = 1; a[1] = 2; a[2] = 3;
%t A298370 a[n_] := a[n] = a[n - 1] + a[n - 2] + Sum[k*a[Floor[n/k]], {k, 2, n}];
%t A298370 Table[a[n], {n, 0, 30}]  (* A298370 *)
%o A298370 (Python)
%o A298370 from functools import lru_cache
%o A298370 @lru_cache(maxsize=None)
%o A298370 def A298370(n):
%o A298370     if n <= 2:
%o A298370         return n+1
%o A298370     c, j = A298370(n-1)+A298370(n-2), 2
%o A298370     k1 = n//j
%o A298370     while k1 > 1:
%o A298370         j2 = n//k1 + 1
%o A298370         c += (j2*(j2-1)-j*(j-1))*A298370(k1)//2
%o A298370         j, k1 = j2, n//j2
%o A298370     return c+2*(n*(n+1)-j*(j-1))//2 # _Chai Wah Wu_, Mar 31 2021
%Y A298370 Cf. A001622, A000045, A298338.
%K A298370 nonn,easy
%O A298370 0,2
%A A298370 _Clark Kimberling_, Feb 10 2018