

A005536


a(0) = 0, a(2n) = n  a(n)  a(n1), 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
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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. 569578 of C. J. NashWilliams 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 kregular Sequences, II, Theoret. Computer Sci., 307 (2003), 329.
HsienKuei 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.
HsienKuei Hwang, S. Janson, T. H. Tsai, 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.
R. Stephan, Some divideandconquer 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/(1x)^2) * Sum_{k>=0} (1)^k*x^2^k/(1 + x^2^k).  Ralf Stephan, Apr 27 2003
a(n) = n*(n2) + 3*Sum_{k=1..n1} 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*(n2)+3*sum(k=1, n1, 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/(1x)^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



