

A001057


Canonical enumeration of integers: interleaved positive and negative integers with zero prepended.


101



0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31
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OFFSET

0,4


COMMENTS

Unsigned sequence (A008619) gives number of partitions of n in which the greatest part is 2.  Robert G. Wilson v, Jan 11 2002
Go forwards and backwards with increasing step sizes.  Daniele Parisse and Franco Virga (daniele.parisse(AT)eads.com), Jun 06 2005
The partial sums of the divergent series 1  2 + 3  4 + ... give this sequence. Euler summed it to 1/4 which was one of the first examples of summing divergent series.  Michael Somos, May 22 2007
From Peter Luschny, Jul 12 2009: (Start)
The general formula for alternating sums of powers is in terms of the SwissKnife polynomials P(n,x) A153641 2^(n1)(P(n,1)(1)^k P(n,2k+1)). Thus
a(k) = 2^(2)(P(1,1)(1)^k P(1,2k+1)). (End)
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=1, A[i,i1]=1, and A[i,j]=0 otherwise. Then, for n>=4, a(n3)=(1)^(n1)*coeff(charpoly(A,x),x).  Milan Janjic, Jan 26 2010
Cantor ordering of the integers producing a 11 and onto correspondence between the natural numbers and the integers showing that the set Z of integers has the same cardinality as the set N of natural numbers. The cardinal of N is the first transfinite cardinal aleph_null (or aleph_naught), which is the cardinality of a given infinite set if and only it is countably infinite (denumerable), i.e., it can be put in 11 and onto correspondence (with a proper Cantor ordering) with the natural numbers.  Daniel Forgues, Jan 23 2010
a(n) is the determinant of the (n+2) X (n+2) (0,1)Toeplitz matrix M satisfying: M(i,j)=0 iff i=j or i=j1. The matrix M arises in the variation of ménage problem where not a round table, but one side of a rectangular table is considered (see comments of Vladimir Shevelev in A000271). Namely M(i,j) defines the class of permutations p of 1,2,...,n+2 such that p(i)<>i and p(i)<>i+1 for i=1,2,...,n+1, and p(n+2)<>n+2. And a(n) is also the difference between the number of even and odd such permutations.  Dmitry Efimov, Mar 02 2017


LINKS

T. D. Noe, Table of n, a(n) for n = 0..1000
D. Efimov, Determinants of generalized binary band matrices, arXiv:1702.05655 [math.RA], 2017.
G. Myerson and A. J. van der Poorten, Some problems concerning recurrence sequences, Amer. Math. Monthly 102 (1995), no. 8, 698705.
Omar E. Pol, Illustration of initial terms of A001057, A005132, A000217
Wikipedia, 1  2 + 3  4 + ...
Index entries for linear recurrences with constant coefficients, signature (1,1,1).


FORMULA

Euler transform of [1, 2] is sequence a(n+1).  Michael Somos, Jun 11 2003
G.f.: x / ((1 + x) * (1  x^2)).  Michael Somos, Jul 20 1999
E.g.f.: (exp(x)  (1  2*x) * exp(x)) / 4.  Michael Somos, Jun 11 2003
a(n) = 1  2*a(n1) a(n2); a(2*n) = n, a(2*n+1) = n+1.  Michael Somos, Jul 20 1999
a(n+1) = A008619(n). a(n1) = A004526(n).  Michael Somos, Jul 20 1999
a(n) = a(n1) + a(n2) + a(n3). a(n) = (1)^(n+1) * floor((n+1) / 2).  Michael Somos, Jun 11 2003
a(1) = 1, a(n) = a(n1)+n or a(n1)n whichever is closer to 0 on the number line. Or abs(a(n)) = min{abs(a(n1)+n), abs(a(n1)n)}.  Amarnath Murthy, Jul 01 2003
a(n) = Sum_{k=0..n} k*(1)^(k+1).  Paul Barry, Aug 20 2003
a(n) = (1(2n+1)*(1)^n)/4.  Paul Barry, Feb 02 2004
a(0) = 0; a(n) = (1)^(n1) * (na(n1)) for n >= 1.  Rick L. Shepherd, Jul 14 2004
a(n) = a(n1)n*(1)^n, a(0)=0; or a(n) = a(n1)+(1(1)^n)/2, a(0)=0.  Daniele Parisse and Franco Virga (daniele.parisse(AT)eads.com), Jun 06 2005
a(n) = ceiling(n/2) * (1)^(n+1), n >= 0.  Franklin T. AdamsWatters, Nov 25 2011 (corrected by Daniel Forgues, Jul 21 2012)
a(n) = a(1n) for all n in Z.  Michael Somos, Jun 05 2013
Sum_{n>=1} 1/a(n) = 0.  Jaume Oliver Lafont, Jul 14 2017


EXAMPLE

G.f. = x  x^2 + 2*x^3  2*x^4 + 3*x^5  3*x^6 + 4*x^7  4*x^8 + 5*x^9  5*x^10 + ...


MAPLE

a := n > (1(1)^n*(2*n+1))/4; # Peter Luschny, Jul 12 2009


MATHEMATICA

Join[{0}, Riffle[Range[35], Range[35]]] (* Harvey P. Dale, Sep 21 2011 *)
a[ n_] := (1)^n Ceiling[n/2]; (* Michael Somos, Jun 05 2013 *)
LinearRecurrence[{1, 1, 1}, {0, 1, 1}, 63] (* JeanFrançois Alcover, Jan 07 2019 *)


PROG

(PARI) {a(n) = if( n%2, n\2 + 1, n/2)}; /* Michael Somos, Jul 20 1999 */
(Haskell)
a001057 n = (n' + m) * (1) ^ (1  m) where (n', m) = divMod n 2
a001057_list = 0 : concatMap (\x > [x, x]) [1..]
 Reinhard Zumkeller, Apr 02 2012


CROSSREFS

Cf. A008619, A004526, A166711, A166871, A130472 (negation), A142150 (partial sums), A010551 (partial products for n > 0).
Alternating row sums of A104578 are a(n+1), for n >= 0.
Sequence in context: A244325 A168050 A065033 * A127365 A130472 A076938
Adjacent sequences: A001054 A001055 A001056 * A001058 A001059 A001060


KEYWORD

sign,nice,core,easy


AUTHOR

N. J. A. Sloane


EXTENSIONS

Thanks to Michael Somos for helpful comments.
Name edited by Franklin T. AdamsWatters, Jan 30 2012


STATUS

approved



