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A131576 Number of ways to represent n as sum of even number of consecutive integers. 13
0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 2, 1, 1, 0, 1, 1, 2, 0, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 2, 1, 1, 0, 1, 1, 2, 1, 1, 1, 2, 0, 2, 1, 1, 1, 1, 1, 2, 0, 2, 1, 1, 1, 2, 1, 1, 0, 1, 1, 3, 1, 1, 2, 1, 0, 2, 1, 1, 1, 2, 1, 2, 0, 1, 2, 1, 1, 2, 1, 2, 0, 1, 1, 2, 1, 1, 2, 1, 0, 4 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,21

COMMENTS

Number of odd divisors of n greater than sqrt(2*n).

Conjecture: a(n) is also the number of pairs of equidistant subparts in the symmetric representation of sigma(n). (Cf. A279387). - Omar E. Pol, Feb 22 2017

Indices of nonzero terms give A281005. - Omar E. Pol, Mar 04 2018

LINKS

Table of n, a(n) for n=1..105.

M. D. Hirschhorn and P. M. Hirschhorn, Partitions into Consecutive Parts, Mathematics Magazine: 2003, Volume 76, Number 4, Pages: 306-308.

FORMULA

G.f.: Sum_{k>1} x^(k*(2*k+1))/(1-x^(2*k)).

a(A000040(i))=1 for i=1,2,3,... a(A000079(j))=0 for j=0,1,2,3,... - R. J. Mathar, Sep 13 2007

Conjectures: a(n) = (A001227(n) - A067742(n))/2 = A001227(n) - A082647(n) = A082647(n) - A067742(n). - Omar E. Pol, Feb 22 2017

EXAMPLE

a(11)=1 because we have 11=5+6; a(21)=2 because we have 21=10+11=1+2+3+4+5+6; a(75)=3 because we have 75=37+38=10+11+12+13+14+15=3+4+5+6+7+8+9+10+11+12.

MAPLE

G:=sum(x^(k*(2*k+1))/(1-x^(2*k)), k=1..10): Gser:=series(G, x=0, 85): seq(coeff(Gser, x, n), n=1..80); # Emeric Deutsch, Sep 08 2007

A131576 := proc(n) local dvs, a, k, r; dvs := numtheory[divisors](n) ; a := 0 ; for k in dvs do r := n/k+1 ; if r mod 2 = 0 then if r/2-k >= 1 then a := a+1 ; fi ; fi ; od: RETURN(a) ; end: seq(A131576(n), n=1..120) ; # R. J. Mathar, Sep 13 2007

MATHEMATICA

With[{m = 105}, Rest@ CoefficientList[Series[Sum[x^(k (2 k + 1))/(1 - x^(2 k)), {k, m}], {x, 0, m}], x]] (* Michael De Vlieger, Mar 04 2018 *)

CROSSREFS

Cf. A082647, A001227, A237593.

Sequence in context: A302234 A026920 A060763 * A100073 A257988 A075685

Adjacent sequences:  A131573 A131574 A131575 * A131577 A131578 A131579

KEYWORD

easy,nonn

AUTHOR

Vladeta Jovovic, Aug 28 2007, Sep 16 2007

STATUS

approved

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Last modified August 19 21:24 EDT 2018. Contains 313896 sequences. (Running on oeis4.)