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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 (list; graph; refs; listen; history; text; internal format)
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
Hsien-Kuei Hwang, Svante Janson, Tsung-Hsi 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.
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/(1-x)^2 * ((x(1+x)/(1-x) - Sum_{k>=0} x^2^k/(1-x^2^k))). - Ralf Stephan, Apr 12 2002
a(0) = 0, a(2*n) = a(n) + a(n-1) + 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 <= 4^2-3 = 13 gives 1, 2, 2, 3, 3, 3, 2, 3, 3, 3, 2, 2, 1.
Fr(2, n) for 1 <= n <= 4^3-3 = 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.
MAPLE
a:= proc(n) option remember; `if`(n=0, 0,
a(n-1)-1+add(i, i=Bits[Split](n)))
end:
seq(a(n), n=1..68); # Alois P. Heinz, Feb 03 2024
MATHEMATICA
Accumulate@Table[DigitCount[n, 2, 1] - 1, {n, 68}] (* Ivan Neretin, Sep 07 2017 *)
PROG
(PARI) a(n)=n^2-sum(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
Equals (1/2) * A076178(n).
Sequence in context: A255347 A029910 A063677 * A296206 A079228 A067535
KEYWORD
nonn
AUTHOR
Benoit Cloitre, Dec 12 2002
STATUS
approved

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Last modified April 17 02:36 EDT 2024. Contains 371756 sequences. (Running on oeis4.)