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A298408
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a(n) = 2*a(n-1) - a(n-3) + 2*a(floor(n/2)) + 3*a(floor(n/3)) + ... + n*a(floor(n/n)), where a(0) = 1, a(1) = 1, a(2) = 1.
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3
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1, 1, 1, 6, 20, 53, 130, 277, 574, 1115, 2126, 3862, 7021, 12341, 21553, 36957, 63111, 106224, 178407, 296638, 492231, 811731, 1335994, 2188950, 3583027, 5847108, 9532980, 15512342, 25226123, 40967842, 66506422, 107869832, 174908573, 283452771, 459264017
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internal format)
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OFFSET
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0,4
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COMMENTS
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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.
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LINKS
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MATHEMATICA
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a[0] = 1; a[1] = 1; a[2] = 1;
a[n_] := a[n] = 2*a[n - 1] - a[n - 3] + Sum[k*a[Floor[n/k]], {k, 2, n}];
Table[a[n], {n, 0, 90}] (* A298408 *)
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PROG
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(Python)
from functools import lru_cache
@lru_cache(maxsize=None)
if n <= 2:
return 1
k1 = n//j
while k1 > 1:
j2 = n//k1 + 1
c += (j2*(j2-1)-j*(j-1))*A298408(k1)//2
j, k1 = j2, n//j2
return c+(n*(n+1)-j*(j-1))//2 # Chai Wah Wu, Mar 31 2021
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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