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A054844 Number of ways to write n as the sum of any number of consecutive integers (including the trivial one-term sum n = n). 12
2, 2, 4, 2, 4, 4, 4, 2, 6, 4, 4, 4, 4, 4, 8, 2, 4, 6, 4, 4, 8, 4, 4, 4, 6, 4, 8, 4, 4, 8, 4, 2, 8, 4, 8, 6, 4, 4, 8, 4, 4, 8, 4, 4, 12, 4, 4, 4, 6, 6, 8, 4, 4, 8, 8, 4, 8, 4, 4, 8, 4, 4, 12, 2, 8, 8, 4, 4, 8, 8, 4, 6, 4, 4, 12, 4, 8, 8, 4, 4, 10, 4, 4, 8, 8, 4, 8, 4, 4, 12, 8, 4, 8, 4, 8, 4, 4, 6, 12, 6 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

a(n) = twice the number of odd divisors of n. That is, if d is the divisor function and q is the exponent of the largest power of 2 dividing n, then the a(n) equals 2*d(n)/(q+1). - Andrew Niedermaier, Jul 20 2003

Moebius transform is period 2 sequence [2, 0, ...]. - Michael Somos, Sep 20 2005

a(n) is twice the number of partitions of n into consecutive parts. - Omar E. Pol, Nov 28 2020

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

FORMULA

a(n) = 2*A001227(n). - Andrew Niedermaier, Jul 20 2003

G.f.: Sum_{k>0} 2x^k/(1-x^(2k)) = Sum_{k>0} 2x^(2k-1)/(1-x^(2k-1)). - Michael Somos, Sep 20 2005

a(n) = A010054(n) + A335616(n). - Omar E. Pol, Nov 28 2020

EXAMPLE

a(3) = 4 because 3 = (-2)+(-1)+0+1+2+3 or 0+1+2 or 1+2 or 3; a(13) = 4 because 13 = (-12)+...+13 or (-5)+...+7 or 6+7 or 13.

From Omar E. Pol, Nov 28 2020: (Start)

Illustration of initial terms:

                                        Diagram

n   a(n)                                  _ _

1     2                                 _|1 1|_

2     2                               _|1 _ _ 1|_

3     4                             _|1  |1 1|  1|_

4     2                           _|1   _|   |_   1|_

5     4                         _|1    |1 _ _ 1|    1|_

6     4                       _|1     _| |1 1| |_     1|_

7     4                     _|1      |1  |   |  1|      1|_

8     2                   _|1       _|  _|   |_  |_       1|_

9     6                 _|1        |1  |1 _ _ 1|  1|        1|_

10    4               _|1         _|   | |1 1| |   |_         1|_

11    4             _|1          |1   _| |   | |_   1|          1|_

12    4           _|1           _|   |1  |   |  1|   |_           1|_

13    4         _|1            |1    |  _|   |_  |    1|            1|_

14    4       _|1             _|    _| |1 _ _ 1| |_    |_             1|_

15    8     _|1              |1    |1  | |1 1| |  1|    1|              1|_

16    2    |1                |     |   | |   | |   |     |                1|

...

a(n) is the number of horizontal toothpicks in the n-th level of the diagram. (End)

PROG

(PARI) a(n)=2*sumdiv(n, d, d%2)

(PARI) A054844(n) = (2*numdiv(n>>valuation(n, 2))); \\ Antti Karttunen, Sep 27 2018

CROSSREFS

Cf. A001227, A010054, A054843, A237593, A335616.

Sequence in context: A214212 A100008 A102763 * A057936 A033097 A036845

Adjacent sequences:  A054841 A054842 A054843 * A054845 A054846 A054847

KEYWORD

easy,nonn

AUTHOR

Henry Bottomley, Apr 13 2000

EXTENSIONS

Corrected and extended by Michael Somos, Apr 26 2000

STATUS

approved

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Last modified April 15 00:44 EDT 2021. Contains 342971 sequences. (Running on oeis4.)