A166711 -id:A166711 - cp's OEIS frontend

A001057 Canonical enumeration of integers: interleaved positive and negative integers with zero prepended.

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

Offset: 0

N. J. A. Sloane

Go forwards and backwards with increasing step sizes. - Daniele Parisse and Franco Virga, 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 Swiss-Knife polynomials P(n,x) A153641 2^(-n-1)(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,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=4, a(n-3)=(-1)^(n-1)*coeff(charpoly(A,x),x). - Milan Janjic, Jan 26 2010
Cantor ordering of the integers producing a 1-1 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 if it is countably infinite (denumerable), i.e., it can be put in 1-1 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=j-1. 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

			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 + ...

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, 698-705.
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).

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.

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

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

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] (* Jean-François Alcover, Jan 07 2019 *)

{a(n) = if( n%2, n\2 + 1, -n/2)}; /* Michael Somos, Jul 20 1999 */

def a(n): return n//2 + 1 if n%2 else -n//2
print([a(n) for n in range(63)]) # Michael S. Branicky, Jul 14 2022

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(n-1) -a(n-2); a(2*n) = -n, a(2*n+1) = n+1. - Michael Somos, Jul 20 1999
|a(n+1)| = A008619(n). |a(n-1)| = A004526(n). - Michael Somos, Jul 20 1999
a(n) = -a(n-1) + a(n-2) + a(n-3). a(n) = (-1)^(n+1) * floor((n+1) / 2). - Michael Somos, Jun 11 2003
a(1) = 1, a(n) = a(n-1)+n or a(n-1)-n whichever is closer to 0 on the number line. Or abs(a(n)) = min{abs(a(n-1)+n), abs(a(n-1)-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)^(n-1) * (n-|a(n-1)|) for n >= 1. - Rick L. Shepherd, Jul 14 2004
a(n) = a(n-1)-n*(-1)^n, a(0)=0; or a(n) = -a(n-1)+(1-(-1)^n)/2, a(0)=0. - Daniele Parisse and Franco Virga, Jun 06 2005
a(n) = ceiling(n/2) * (-1)^(n+1), n >= 0. - Franklin T. Adams-Watters, Nov 25 2011 (corrected by Daniel Forgues, Jul 21 2012)
a(n) = a(-1-n) for all n in Z. - Michael Somos, Jun 05 2013
Sum_{n>=1} 1/a(n) = 0. - Jaume Oliver Lafont, Jul 14 2017

Thanks to Michael Somos for helpful comments.

Name edited by Franklin T. Adams-Watters, Jan 30 2012

A009947 Sequence of nonnegative integers, but insert n/2 after every even number n.

Original entry on oeis.org

0, 0, 1, 2, 1, 3, 4, 2, 5, 6, 3, 7, 8, 4, 9, 10, 5, 11, 12, 6, 13, 14, 7, 15, 16, 8, 17, 18, 9, 19, 20, 10, 21, 22, 11, 23, 24, 12, 25, 26, 13, 27, 28, 14, 29, 30, 15, 31, 32, 16, 33, 34, 17, 35, 36, 18, 37, 38, 19, 39, 40, 20, 41, 42, 21, 43, 44, 22, 45, 46

Offset: 0

Bill Gosper

Coefficients in expansion of e/3 = Sum_{n>=1} a(n)/n!, using greedy algorithm.

Numerators of Peirce sequence of order 2.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, Reading, MA, 2nd ed. 1998, p. 151.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

Cf. A071281, A214090 (parity), A001477.

Cf. A166711 (signed).

a009947 n = a009947_list !! n
a009947_list = concatMap (\x -> [2 * x, x, 2 * x + 1]) [0..]
-- Reinhard Zumkeller, Jul 06 2012

A009947 := proc(a,n) local i,b,c; b := a; c := [ floor(b) ]; for i from 1 to n-1 do b := b-c[ i ]/i!; c := [ op(c), floor(b*(i+1)!) ]; od; c; end:

Flatten[Table[If[EvenQ[n],{n,n/2},n],{n,0,40}]] (* Harvey P. Dale, Feb 17 2016 *)

a(n)=if(n%3==1, n\3, n\3*2+!!(n%3)) \\ Charles R Greathouse IV, Sep 02 2015

concat(vector(2), Vec(x^2*(x^3+x^2+2*x+1) / ((x-1)^2*(x^2+x+1)^2) + O(x^100))) \\ Colin Barker, Mar 29 2017

G.f.: x^2*(x^3+x^2+2*x+1) / ((x-1)^2*(x^2+x+1)^2). - Colin Barker, Aug 31 2013
a(n) = (n^2-n+floor(n/3)*(18*floor(n/3)^2-3*(4*n-7)*floor(n/3)+2*n^2-10*n+7))/2. - Luce ETIENNE, Mar 29 2017
Sum_{n>=2} (-1)^n/a(n) = Pi/4 + log(2)/2. - Amiram Eldar, Jan 11 2023

A166871 Permutation of the integers: 3 positives, 2 negatives.

Original entry on oeis.org

0, 1, 2, 3, -1, -2, 4, 5, 6, -3, -4, 7, 8, 9, -5, -6, 10, 11, 12, -7, -8, 13, 14, 15, -9, -10, 16, 17, 18, -11, -12, 19, 20, 21, -13, -14, 22, 23, 24, -15, -16, 25, 26, 27, -17, -18, 28, 29, 30, -19, -20, 31, 32, 33, -21, -22, 34, 35, 36, -23, -24, 37, 38, 39, -25, -26, 40, 41

Offset: 0

Jaume Oliver Lafont, Oct 22 2009

This sequence enumerates the denominators with sign in case p=3 and n=2 of:
log(p/n) = sum( i>=0, sum(p*i+1<=j<=p*(i+1),1/j) - sum(n*i+1<=j<=n*(i+1),1/j) )
Similar sequences can be constructed for the logarithm of any rational r=p/n (p,n>0), enumerating p positive integers and n negative integers every p+n terms.
Case p=2, n=1 is A166711.
Case p=1, n=1 is A001057.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Wikipedia, Riemann series theorem
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,2,0,0,0,0,-1)

Cf. A016578, A166711.

LinearRecurrence[{0,0,0,0,2,0,0,0,0,-1},{0,1,2,3,-1,-2,4,5,6,-3}, 100] (* G. C. Greubel, May 27 2016 *)

Sum_{k>0} 1/a(k) = log(3/2).

G.f.: x*(1+2*x+3*x^2-x^3-2*x^4+2*x^5+x^6-x^8)/((x-1)^2*(x^4+x^3+x^2+x+1)^2 ).

keyword frac removed Jaume Oliver Lafont, Nov 02 2009

A167407 T(m,n) is -m if n=0, 1 elsewhere.

Original entry on oeis.org

0, -1, 1, -2, 1, 1, -3, 1, 1, 1, -4, 1, 1, 1, 1, -5, 1, 1, 1, 1, 1, -6, 1, 1, 1, 1, 1, 1, -7, 1, 1, 1, 1, 1, 1, 1, -8, 1, 1, 1, 1, 1, 1, 1, 1, -9, 1, 1, 1, 1, 1, 1, 1, 1, 1, -10, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -12, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -13, 1, 1, 1, 1, 1

Offset: 0

Jaume Oliver Lafont, Nov 03 2009, Nov 04 2009

This triangle encodes a family of conditionally convergent series for the logarithm of positive integers, according to: log(m)=Sum_{n>0} T(m-1,n mod m)/n.
The second row of the triangle, m=1, corresponds to Mercator's series:
  log(2)=1-1/2+1/3-1/4+1/5-1/6+-...

			Triangle begins:
   0;
  -1,1;
  -2,1,1;
  -3,1,1,1;
  -4,1,1,1,1;
  ...

Cf. A061347, A166711, A166871.

Flatten[Table[{-n,Table[1,{n}]},{n,0,15}]] (* Harvey P. Dale, Apr 17 2015 *)

Revised by Jaume Oliver Lafont, Nov 11 2009

A359027 A line of empty cells is filled by successive terms t >= 1 with t+1 copies of t and gaps of t empty cells between them.

Original entry on oeis.org

1, 2, 1, 3, 4, 2, 5, 6, 2, 3, 7, 8, 4, 3, 9, 10, 5, 3, 11, 4, 12, 6, 13, 14, 4, 5, 7, 15, 16, 4, 8, 6, 5, 17, 18, 9, 19, 7, 5, 20, 6, 10, 21, 22, 5, 8, 11, 23, 6, 7, 24, 12, 9, 25, 26, 6, 13, 8, 7, 27, 10, 28, 6, 14, 29, 30, 9, 7, 11, 8, 15, 31, 32, 16, 12, 7, 10

Offset: 1

Tamas Sandor Nagy, Dec 12 2022

We write 1 into the first cell, then by leaving a gap of one empty cell we write another 1 into the third cell.
Next, we write 2 into the first available empty cell, then write two more 2's by leaving gaps of two empty cells between them. And so on.
It appears that the absolute values of A166711 appear in order nicely embedded into this sequence. - Thomas Scheuerle, Dec 12 2022

			Cell filling begins, starting from an empty line:
  | | | | | | | | | | | | | | | | | | |
  .
  |1| |1| | | | | | | | | | | | | | | |
  .
  |1|2|1| | |2| | |2| | | | | | | | | |
  .
  |1|2|1|3| |2| | |2|3| | | |3| | | |3|

Thomas Scheuerle, Scatter plot Y: ((1/2)+a(n))^(4/3) X: n = 1...20000

Cf. A166711.

Cf. A028920 (with infinite copies).

function a = A359027( max_n )
    a = zeros(1,max_n);
    f = 1:max_n; k = 1;
    while ~isempty(f)
        j = f(1:(k+1):end);
        a(j(j(1:min(k+1,length(j))) <= max_n )) = k;
        k = k+1;
        f = find(a == 0);
    end
end % Thomas Scheuerle, Dec 12 2022

cp's OEIS Frontend

A001057 Canonical enumeration of integers: interleaved positive and negative integers with zero prepended.

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A009947 Sequence of nonnegative integers, but insert n/2 after every even number n.

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A166871 Permutation of the integers: 3 positives, 2 negatives.

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A167407 T(m,n) is -m if n=0, 1 elsewhere.

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A359027 A line of empty cells is filled by successive terms t >= 1 with t+1 copies of t and gaps of t empty cells between them.

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