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A060973
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a(2*n+1) = a(n+1)+a(n), a(2*n) = 2*a(n), with a(1)=0 and a(2)=1.
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5
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0, 1, 1, 2, 2, 2, 3, 4, 4, 4, 4, 4, 5, 6, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 10, 11, 12, 13, 14, 15, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32
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listen;
history;
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internal format)
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OFFSET
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1,4
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LINKS
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R. J. Mathar, Table of n, a(n) for n = 1..1000
Michael De Vlieger, Log-log scatterplot of a(n), n = 1..2^16.
H.-K. Hwang, S. Janson, and T.-H. Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications. Preprint 2016.
H.-K. Hwang, S. Janson, and T.-H. Tsai, Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications. ACM Transactions on Algorithms, 13:4 (2017), #47. doi:10.1145/3127585
Jeffrey Shallit, Intertwining of Complementary Thue-Morse Factors, arXiv:2203.02917 [cs.FL], 2022.
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions
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FORMULA
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a(n) = n-A006165(n) = A006165(n)-A053646(n) = (n-A053646(n))/2 [for n>1 ]. If n = 2*2^m+k with 0< = k< = 2^m, then a(n) = 2^m; if n = 3*2^m+k with 0< = k< = 2^m, then a(n) = 2^m+k.
G.f. -x/(1-x) + x/(1-x)^2 * (1 + sum(k>=0, t^2(t-1), t=x^2^k)). - Ralf Stephan, Sep 12 2003
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EXAMPLE
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a(6)=2*a(3)=2*1=2. a(7)=a(3)+a(4)=1+2=3.
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MAPLE
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A060973 := proc(n)
option remember;
if n <= 2 then
return n-1;
fi;
if n mod 2 = 0 then
2*procname(n/2)
else
procname((n-1)/2)+procname((n+1)/2);
fi;
end proc:
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MATHEMATICA
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nn = 77; Array[Set[a[#], # - 1] &, 2]; Do[Set[a[i], If[EvenQ[i], 2 a[i/2], a[# + 1] + a[#] &[(i - 1)/2]]], {i, 3, nn}]; Array[a, nn] (* Michael De Vlieger, Mar 22 2022 *)
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PROG
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(Python)
from functools import lru_cache
@lru_cache(maxsize=None)
def A060973(n): return n-1 if n <= 2 else A060973(n//2) + A060973((n+1)//2) # Chai Wah Wu, Mar 08 2022
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CROSSREFS
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Cf. A006165, A053646.
Sequence in context: A228482 A091822 A173022 * A352228 A097915 A255072
Adjacent sequences: A060970 A060971 A060972 * A060974 A060975 A060976
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KEYWORD
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nonn
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AUTHOR
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Henry Bottomley, May 09 2001
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STATUS
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approved
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