cp's OEIS Frontend

A054844 Number of ways to write n as the sum of any number of consecutive integers (including the trivial one-term sum n = n).

%I A054844 #33 Nov 30 2020 08:45:31
%S A054844 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,
%T A054844 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,
%U A054844 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
%N A054844 Number of ways to write n as the sum of any number of consecutive integers (including the trivial one-term sum n = n).
%C A054844 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
%C A054844 Moebius transform is period 2 sequence [2, 0, ...]. - _Michael Somos_, Sep 20 2005
%C A054844 a(n) is twice the number of partitions of n into consecutive parts. - _Omar E. Pol_, Nov 28 2020
%H A054844 Antti Karttunen, <a href="/A054844/b054844.txt">Table of n, a(n) for n = 1..65537</a>
%F A054844 a(n) = 2*A001227(n). - _Andrew Niedermaier_, Jul 20 2003
%F A054844 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
%F A054844 a(n) = A010054(n) + A335616(n). - _Omar E. Pol_, Nov 28 2020
%e A054844 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.
%e A054844 From _Omar E. Pol_, Nov 28 2020: (Start)
%e A054844 Illustration of initial terms:
%e A054844                                         Diagram
%e A054844 n   a(n)                                  _ _
%e A054844 1     2                                 _|1 1|_
%e A054844 2     2                               _|1 _ _ 1|_
%e A054844 3     4                             _|1  |1 1|  1|_
%e A054844 4     2                           _|1   _|   |_   1|_
%e A054844 5     4                         _|1    |1 _ _ 1|    1|_
%e A054844 6     4                       _|1     _| |1 1| |_     1|_
%e A054844 7     4                     _|1      |1  |   |  1|      1|_
%e A054844 8     2                   _|1       _|  _|   |_  |_       1|_
%e A054844 9     6                 _|1        |1  |1 _ _ 1|  1|        1|_
%e A054844 10    4               _|1         _|   | |1 1| |   |_         1|_
%e A054844 11    4             _|1          |1   _| |   | |_   1|          1|_
%e A054844 12    4           _|1           _|   |1  |   |  1|   |_           1|_
%e A054844 13    4         _|1            |1    |  _|   |_  |    1|            1|_
%e A054844 14    4       _|1             _|    _| |1 _ _ 1| |_    |_             1|_
%e A054844 15    8     _|1              |1    |1  | |1 1| |  1|    1|              1|_
%e A054844 16    2    |1                |     |   | |   | |   |     |                1|
%e A054844 ...
%e A054844 a(n) is the number of horizontal toothpicks in the n-th level of the diagram. (End)
%o A054844 (PARI) a(n)=2*sumdiv(n,d,d%2)
%o A054844 (PARI) A054844(n) = (2*numdiv(n>>valuation(n, 2))); \\ _Antti Karttunen_, Sep 27 2018
%Y A054844 Cf. A001227, A010054, A054843, A237593, A335616.
%K A054844 easy,nonn
%O A054844 1,1
%A A054844 _Henry Bottomley_, Apr 13 2000
%E A054844 Corrected and extended by _Michael Somos_, Apr 26 2000