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A005536 a(0) = 0, a(2n) = n - a(n) - a(n-1), a(2n+1) = n - 2a(n) + 1.
(Formerly M2274)
5
1, 0, 0, 1, 3, 3, 4, 3, 3, 1, 0, 0, 1, 0, 0, 1, 3, 3, 4, 6, 9, 10, 12, 12, 13, 12, 12, 13, 15, 15, 16, 15, 15, 13, 12, 12, 13, 12, 12, 10, 9, 6, 4, 3, 3, 1, 0, 0, 1, 0, 0, 1, 3, 3, 4, 3, 3, 1, 0, 0, 1, 0, 0, 1, 3, 3, 4, 6, 9, 10, 12, 12, 13, 12, 12, 13, 15, 15, 16, 18, 21, 22, 24, 27, 31, 33 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

A "Von Koch" sequence generated by the first Feigenbaum symbolic sequence.

REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. G. Stanton, W. L. Kocay and P. H. Dirksen, Computation of a combinatorial function, pp. 569-578 of C. J. Nash-Williams and J. Sheehan, editors, Proceedings of the Fifth British Combinatorial Conference 1975. Utilitas Math., Winnipeg, 1976.

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

J.-P. Allouche and J. Shallit, The Ring of k-regular Sequences, II, Theoret. Computer Sci., 307 (2003), 3-29.

Hsien-Kuei Hwang, S. Janson, 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.

Hsien-Kuei Hwang, S. Janson, 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.

R. Stephan, Some divide-and-conquer sequences ...

R. Stephan, Table of generating functions

Index entries for sequences related to binary expansion of n

FORMULA

Partial sums of A065359. a(n) = Sum_{i=0..n} Sum_{k=0..i} (-1)^k*(floor(i/2^k) - 2*floor(i/2^(k+1))). - Benoit Cloitre, Mar 28 2004

G.f. (1/(1-x)^2) * Sum_{k>=0} (-1)^k*x^2^k/(1 + x^2^k). - Ralf Stephan, Apr 27 2003

a(n) = -n*(n-2) + 3*Sum_{k=1..n-1} Sum_{i=1..k} A035263(i+1), where A035263 is the first Feigenbaum symbolic sequence. - Benoit Cloitre, May 29 2003

MATHEMATICA

a[n_] := a[n] = If[n == 0, 0, hn = Floor[n/2]; If[OddQ[n], hn - 2 a[hn] + 1, hn - a[hn] - a[hn - 1]]]; t = Table[a[n], {n, 100}] (* T. D. Noe, Mar 22 2012 *)

PROG

(PARI) a(n)=-n*(n-2)+3*sum(k=1, n-1, sum(i=1, k, abs(subst(Pol(binary(i+1))- Pol(binary(i)), x, 1)%2))) \\ Benoit Cloitre, May 29 2003

(PARI) a(n)=polcoeff(1/(1-x)^2*sum(k=0, 10, (-1)^k*x^2^k/(1+x^2^k)) +O(x^(n+1)), n)

CROSSREFS

Cf. A071992, A073059.

Sequence in context: A023647 A239207 A082978 * A172515 A188590 A080038

Adjacent sequences:  A005533 A005534 A005535 * A005537 A005538 A005539

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms and better description from Ralf Stephan, Sep 13 2003

STATUS

approved

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Last modified October 19 17:53 EDT 2018. Contains 316376 sequences. (Running on oeis4.)