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A298406
a(n) = 2*a(n-1) - a(n-3) + a(floor(n/2)) + a(floor(n/3)) + ... + a(floor(n/n)), where a(0) = 1, a(1) = 1, a(2) = 1.
3
1, 1, 1, 3, 8, 19, 42, 84, 163, 301, 547, 961, 1682, 2879, 4902, 8241, 13807, 22917, 37962, 62487, 102690, 168096, 274798, 448000, 729829, 1186797, 1928729, 3130905, 5080360, 8237339, 13352743, 21634097, 35045477, 56753250, 91896553, 148771833, 240830555
OFFSET
0,4
COMMENTS
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.
LINKS
MATHEMATICA
a[0] = 1; a[1] = 1; a[2] = 1;
a[n_] := a[n] = 2*a[n - 1] - a[n - 3] + Sum[a[Floor[n/k]], {k, 2, n}];
Table[a[n], {n, 0, 90}] (* A298406 *)
PROG
(Python)
from functools import lru_cache
@lru_cache(maxsize=None)
def A298406(n):
if n <= 2:
return 1
c, j = 2*A298406(n-1)-A298406(n-3), 2
k1 = n//j
while k1 > 1:
j2 = n//k1 + 1
c += (j2-j)*A298406(k1)
j, k1 = j2, n//j2
return c+n-j+1 # Chai Wah Wu, Mar 31 2021
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Feb 10 2018
STATUS
approved