A005590 a(0) = 0, a(1) = 1, a(2n) = a(n), a(2n+1) = a(n+1) - a(n). (Formerly M0048)

0, 1, 1, 0, 1, -1, 0, 1, 1, -2, -1, 1, 0, 1, 1, 0, 1, -3, -2, 1, -1, 2, 1, -1, 0, 1, 1, 0, 1, -1, 0, 1, 1, -4, -3, 1, -2, 3, 1, -2, -1, 3, 2, -1, 1, -2, -1, 1, 0, 1, 1, 0, 1, -1, 0, 1, 1, -2, -1, 1, 0, 1, 1, 0, 1, -5, -4, 1, -3, 4, 1, -3, -2, 5, 3, -2, 1, -3, -2, 1, -1, 4, 3, -1, 2, -3, -1, 2, 1, -3, -2, 1, -1, 2, 1, -1, 0, 1, 1, 0, 1, -1, 0, 1, 1

Offset

0,10

Comments

If "-" in the definition is changed to "+", we get Stern's diatomic sequence A002487.

Sequence is 2-regular.

Let M = a triangular matrix with (1, 1, -1, 0, 0, 0, ...) in every column >k=1 shifted down twice from the previous column. Then A005590 starting with 1 = lim_{n->infinity} M^n, the left-shifted vector considered as a sequence. - Gary W. Adamson, Apr 13 2010

a(A001969(n)) <= 0; a(A000069(n)) > 0. - Reinhard Zumkeller, Apr 11 2012

References

B. Reznick, A new sequence with many properties, Abstract 809-10-185, Abstracts Amer. Math. Soc., 5 (1984), p. 16. [See link below]

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

Links

T. D. Noe, Table of n, a(n) for n = 0..10000

J.-P. Allouche and M. Mendes France, Stern-Brocot polynomials and power series, arXiv preprint arXiv:1202.0211 [math.NT], 2012.

J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197.

Michael Gilleland, Some Self-Similar Integer Sequences

Bruce Reznick, Some extremal problems for continued fractions, Ill. J. Math., 29 (1985), 261-279.

Bruce Reznick, Letter to N. J. A. Sloane, Jun 03 1991; also annotated scanned copy of B. Reznick, A new sequence with many properties, Abstract 809-10-185, Abstracts Amer. Math. Soc., 5 (1984), p. 16.

Ralf Stephan, Divide-and-conquer generating functions. I. Elementary sequences, arXiv:math/0307027 [math.CO], 2003.

Formula

G.f.: x*Product_{k>=0} (1+x^(2^k) - x^2^(k+1)). - Ralf Stephan, Apr 26 2003

Conjecture: a(3n)=0 iff n in A003714. - Ralf Stephan, May 02 2003

a(n) = Sum_{k=0..n-1} (-1)^A010060(n-k-1)*(binomial(k, n-k-1) mod 2). - Paul Barry, Mar 26 2005

G.f. satisfies A(x) = (1 + 1/x - x) * A(x^2). - Michael Somos, Sep 17 2003

limsup log(|a(n)|)/(log n) = 0.4309... [Reznick] - N. J. A. Sloane, Jul 23 2016

From Chai Wah Wu, Dec 20 2016: (Start)

a(2^k*n+1) = a(n+1) - k*a(n);

a(2^k*n+3) = a(n) for k >= 2;

a(2^k*n+5) = -a(2^(k-1)*n+1) for k >= 3;

a(2^k*n+7) = a(2^(k-2)*n+1) for k >= 4;

a(2^k*n+2^k-1) = a(n) if k is even;

a(2^k*n+2^k-1) = a(n+1)-a(n)= a(2*n+1) if k is odd.

This implies that

a(2^k+1) = 1-k;

a(2^k+3) = 1 for k >= 2;

a(2^k+5) = k-2 for k >= 3;

a(2^k+7) = 3-k for k >= 4;

a(2^k-1) = 0 if k is even;

a(2^k-1) = 1 if k is odd.

(End)

Example

G.f. = x + x^2 + x^4 - x^5 + x^7 + x^8 - 2*x^9 - x^10 + x^12 + x^13 + x^14 + ...

Maple

A005590 := proc(n) option remember; if n <= 1 then n; elif n mod 2 = 0 then A005590(n/2); else A005590((n+1)/2)-A005590((n-1)/2); fi; end;

Mathematica

a[0] = 0; a[1] = 1; a[n_] := a[n] = If[OddQ[n], a[(n-1)/2 + 1] - a[(n-1)/2], a[n/2]]; Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Nov 27 2012 *)

Prog

(PARI) {a(n) = if( n<=1, n>0, if(n%2, a(n\2+1) - a(n\2), a(n/2)))}; /* Michael Somos, Sep 17 2003 */
(Haskell)
import Data.List (transpose)
a005590 n = a005590_list !! n
a005590_list = 0 : 1 : concat (tail $ transpose
   [a005590_list, zipWith (-) (tail a005590_list) a005590_list])
-- Reinhard Zumkeller, Apr 11 2012
(Python)
l=[0, 1]
for n in range(2, 101):
    l.append(l[n//2] if n%2==0 else l[(n + 1)//2] - l[(n - 1)//2])
print(l) # Indranil Ghosh, Jun 28 2017

Crossrefs

Cf. A002487, A182093 (partial sums).

Sequence in context: A076453 A263657 A261769 * A142598 A274372 A037800

Adjacent sequences: A005587 A005588 A005589 * A005591 A005592 A005593

Keyword

sign,nice,easy,look

Author

N. J. A. Sloane

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 28 2003

Signs corrected by Ralf Stephan, Apr 26 2003

Status

approved