A158118 - OEIS

%I #26 Dec 13 2014 00:48:39

%S 0,0,0,0,0,0,0,0,0,0,0,2,0,0,4,2,0,0,4,124,0,0,536,712,0,0,4574,2260,

%T 0,0,10634,73758,0,0,406032,638830,0,0,4249160,3263500,0,0,21907736,

%U 82561050,0,0,485798436,945916970,0,0,5968541478,6839493576,0,0

%N Number of solutions of +-1+-2^3+-3^3..+-n^3=0.

%C Constant term in the expansion of (x + 1/x)(x^8 + 1/x^8)..(x^n^3 + 1/x^n^3).

%C a(n) = 0 for any n=1 (mod 4) or n=2 (mod 4).

%C The expansion above and the integral representation formula below are due to Andrica & Tomescu. The asymptotic formula is a conjecture; see Andrica & Ionascu. - _Jonathan Sondow_, Nov 06 2013

%H Ray Chandler, <a href="/A158118/b158118.txt">Table of n, a(n) for n = 1..130</a>

%H D. Andrica and E. J. Ionascu, <a href="http://www.dorinandrica.ro/files/presentation-INTEGERS-2013.pdf">Variations on a result of Erdős and Surányi</a>, INTEGERS 2013 slides.

%H Dorin Andrica and Ioan Tomescu, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL5/Tomescu/tomescu4.html">On an Integer Sequence Related to a Product of Trigonometric Functions, and Its Combinatorial Relevance</a>, J. Integer Sequences, 5 (2002), Article 02.2.4.

%F a(n) = 2 * A113263(n).

%F Integral representation: a(n)=((2^n)/Pi)*int_0^Pi prod_{k=1}^n cos(x*k^3) dx.

%F Asymptotic formula: a(n)=(2^n)*sqrt(14/(Pi*n^7))*(1+o(1)) as n-->infty; n=-1 or 0 (mod 4).

%e Example: For n=12 the a(12) = 2 solutions are:

%e +1+8-27+64-125-216-343+512+729-1000-1331+1728=0,

%e -1-8+27-64+125+216+343-512-729+1000+1331-1728=0.

%p N:=60: p:=1: a:=[]: for n from 1 to N do p:=expand(p*( x^(n^3) + x^(-n^3) )): a:=[op(a), coeff(p,x,0)]: od:a;

%Y Equals twice A113263.

%Y Cf. A063865, A158092, A019568. - _Pietro Majer_, Mar 15 2009

%K nonn

%O 1,12

%A _Pietro Majer_, Mar 12 2009