

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

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


LINKS

Ivan Neretin, Table of n, a(n) for n = 1..10000
HsienKuei Hwang, Svante Janson, and TsungHsi 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.
HsienKuei Hwang, Svante Janson, TsungHsi Tsai, Exact and Asymptotic Solutions of a DivideandConquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms 13:4 (2017), #47.
Ralf Stephan, Some divideandconquer sequences with (relatively) simple generating functions, 2004.
Ralf Stephan, Table of generating functions (ps file).
Ralf Stephan, Table of generating functions (pdf file).


FORMULA

a(n) = n^2  Sum_{k=1..n} A005187(k);
a(n) = n^2  Sum_{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(2*n) = a(n) + a(n1) + n  1, a(2*n+1) = 2*a(n) + n. Also, 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)))
(MAGMA) [n^2(&+[ &+[Valuation(2*v, 2):v in [1..u]]:u in [1..n]]):n in [1..70]]; // Marius A. Burtea, Oct 24 2019


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



