OFFSET
3,1
COMMENTS
Crandall defines phi_2(r_1,r_2) = (1/Pi^2) Sum_{positive & negative odd m_1, m_2} cos(Pi m_1 r_1) cos(Pi m_2 r_2) / (m_1^2+m_2^2).
Lemma: 4a(n) < n^2. Proof: 4a(2) = 2 < 2^2; 4a(4k+1) = 16k^2 < (4k+1)^2; 4a(4k+3) = (4k+2)(4k+4) = (4k+3)^2-1; 4a(p^2 k) = 4p^2 a(pk) < p^2(pk)^2 = (p^2 k)^2; 4 a(jk) = 4 a(j) 4 a(k) < (jk)^2.
Corollary: a(n) <= A198442(n).
REFERENCES
R. Crandall, The Poisson equation and "natural" Madelung constants, preprint 2012 (see section 2 of BBCZ below).
LINKS
Jason Kimberley, Table of n, a(n) for n = 3..10000
D. H. Bailey, J. Borwein, R. Crandall and J. Zucker, Lattice sums arising from the Poisson equation, preprint (2012).
D. H. Bailey, J. M. Borwein and J. S. Kimberley, with an appendix by W. B. Ladd, Computer discovery and analysis of large Poisson polynomials, Experimental Mathematics, August 2016.
J. M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016.
J. M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016. [Cached copy, with permission]
OEIS (Plot 2), Plot of (log n, log a(n))
OEIS (Plot 2), Plot of (n, a(n)) and (n, A198442(n))
FORMULA
a(n) = A079458(n) / 4, for n > 2. - Jason Kimberley, Nov 14 2015
Watson Ladd has proved that the sequence satisfies the following recurrence relations, which were conjectured by Jason Kimberley:
a(1) = 1/4, a(2) = 1/2, for notational convenience;
a(4k+1) = (2k)*(2k) for prime 4k+1;
a(4k+3) = (2k+1)*(2k+2) for prime 4k+3;
a(p^2 k) = p^2 * a(p*k) for prime p;
a(jk) = 4*a(j)*a(k) for j coprime to k.
PROG
(Magma) A218147 := func<n|n eq 2 select 1/2 else IsPrime(n)select
n mod 4 eq 1 select(n div 2)^2 else(n div 2)*(n div 2+1)
else(4^(#fact-1)*&*[p^(2*e-2)*$$(p)where p, e is Explode(p_e):p_e in fact]
where fact is Factorisation(n))>; // Jason Kimberley, Oct 23 2012
(Magma) A218147 := func<n|#UnitGroup(quo<IntegerRing(QuadraticField(-1))|n>)/4>; // Jason Kimberley, Nov 14 2015
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
Jason Kimberley, Oct 21 2012 and Apr 04 2016
EXTENSIONS
Entry revised by N. J. A. Sloane, May 15 2016, to take into account the fact that the conjectured formula for this sequence has now been established by Watson Ladd.
STATUS
approved