A078903 a(n) = n^2 - Sum_{u=1..n} Sum_{v=1..u} valuation(2*v, 2).

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

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

Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi 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.

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.

Ralf Stephan, Some divide-and-conquer 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/(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

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