

A078903


a(n) = n^2  sum(u=1,n,sum(v=1,u,valuation(2*v,2))).


4



0, 0, 1, 1, 2, 3, 5, 5, 6, 7, 9, 10, 12, 14, 17, 17, 18, 19, 21, 22, 24, 26, 29, 30, 32, 34, 37, 39, 42, 45, 49, 49, 50, 51, 53, 54, 56, 58, 61, 62, 64, 66, 69, 71, 74, 77, 81, 82, 84, 86, 89, 91, 94, 97, 101, 103, 106, 109, 113, 116, 120, 124, 129, 129, 130, 131, 133, 134
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OFFSET

1,5


COMMENTS

A fractal generator sequence. Let Fr(m,n) = m*na(n); then the graph of Fr(m,n) for 1<=n<=4^(m+1)3 presents fractal aspects.


REFERENCES

HsienKuei Hwang, S Janson, TH 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; http://140.109.74.92/hk/wpcontent/files/2016/12/aathhrr1.pdf. Also Exact and Asymptotic Solutions of a DivideandConquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585


LINKS

Ivan Neretin, Table of n, a(n) for n = 1..10000
R. Stephan, Some divideandconquer sequences ...
R. Stephan, Table of generating functions


FORMULA

a(n) = n^2sum(k=1, n, A005187(k));
a(n) = n^2sum(u=1, n, sum(v=1, u, A001511(v)));
a(n+1)a(n) = A048881(n).
G.f.: 1/(1x)^2 * ((x(1+x)/(1x)  Sum(k>=0, x^2^k/(1x^2^k)))).  Ralf Stephan, Apr 12 2002
a(0) = 0, a(2n) = a(n) + a(n1) + n  1, a(2n+1) = 2a(n) + n. a(n) = A000788(n)  n.  Ralf Stephan, Oct 05 2003


EXAMPLE

Fr(1, n) for 1<=n<=4^23=13 gives : 1, 2, 2, 3, 3, 3, 2, 3, 3, 3, 2, 2, 1;
Fr(2, n) for 1<=n<=4^33=63 gives : 2, 4, 5, 7, 8, 9, 9, 11, 12, 13, 13, 14, 14, 14, 13, 15, 16, 17, 17, 18, 18, 18, 17, 18, 18, 18, 17, 17, 16, 15, 13, 15, 16, 17, 17, 18, 18, 18, 17, 18, 18, 18, 17, 17, 16, 15, 13, 14, 14, 14, 13, 13, 12, 11, 9, 9, 8, 7, 5, 4, 2


MATHEMATICA

Accumulate@Table[DigitCount[n, 2, 1]  1, {n, 68}] (* Ivan Neretin, Sep 07 2017 *)


PROG

(PARI) a(n)=n^2sum(u=1, n, sum(v=1, u, valuation(2*v, 2)))


CROSSREFS

Cf. A078904, A073504.
Equals 1/2 * A076178(n).
Sequence in context: A255347 A029910 A063677 * A296206 A079228 A067535
Adjacent sequences: A078900 A078901 A078902 * A078904 A078905 A078906


KEYWORD

nonn


AUTHOR

Benoit Cloitre, Dec 12 2002


STATUS

approved



