A088534 Number of representations of n by the quadratic form x^2 + xy + y^2 with 0 <= x <= y.

1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0

Offset: 0

Author

Benoit Cloitre, Nov 16 2003

Keywords

• nonn

Comments

Also, apparently the number of 6-regular plane graphs with n vertices that have only trigonal faces and loops ("({1,3},6)-spheres" from the paper by Michel Deza and Mathieu Dutour Sikiric). - Andrey Zabolotskiy, Dec 22 2021

Examples

			From _M. F. Hasler_, Mar 05 2018: (Start)
a(0) = a(1) = 1 since 0 = 0^2 + 0*0 + 0^2 and 1 = 0^2 + 0*1 + 1^2.
a(2) = 0 since 2 cannot be written as x^2 + xy + y^2.
a(49) = 2 since 49 = 0^2 + 0*7 + 7^2 = 3^2 + 3*5 + 5^2. (End)
		

References

• B. C. Berndt, "On a certain theta-function in a letter of Ramanujan from Fitzroy House", Ganita 43 (1992), 33-43.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
• Michel Deza and Mathieu Dutour Sikiric, Zigzag and Central Circuit Structure of ({1,2,3},6)-spheres, Taiwanese Journal of Mathematics, 16 (June 2012), No. 3, 913-940; see Table 1, p. 916.
• Michel-Marie Deza, Mathieu Dutour Sikiric, and Mikhail Ivanovitch Shtogrin, Geometric Structure of Chemistry-Relevant Graphs, Springer, 2015; see Table 5.4 and Section 5.4.
• Oscar Marmon, Hexagonal Lattice Points on Circles, arXiv:math/0508201 [math.NT], 2005.

Crossrefs

Cf. A003136, A034020, A000196.

Cf. A118886 (indices of values > 1), A198772 (indices of 1's), A198773 (indices of 2's), A198774 (indices of 3's), A198775 (indices of 4's), A198799 (index of 1st term = n).

Cf. A215622.

Programs

Haskell

a088534 n = length
   [(x,y) | y <- [0..a000196 n], x <- [0..y], x^2 + x*y + y^2 == n]
a088534_list = map a088534 [0..]
-- Reinhard Zumkeller, Oct 30 2011

Julia

function A088534(n)
    n % 3 == 2 && return 0
    M = Int(round(2*sqrt(n/3)))
    count = 0
    for y in 0:M, x in 0:y
        n == x^2 + y^2 + x*y && (count += 1)
    end
    return count
end
A088534list(upto) = [A088534(n) for n in 0:upto]
A088534list(104) |> println # Peter Luschny, Mar 17 2018

Mathematica

a[n_] := Sum[Boole[i^2 + i*j + j^2 == n], {i, 0, n}, {j, 0, i}];
Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Jun 20 2018 *)

PARI

a(n)=sum(i=0,n,sum(j=0,i,if(i^2+i*j+j^2-n,0,1)))

PARI

A088534(n,d)=sum(x=0,sqrt(n\3),sum(y=max(x,sqrtint(n-x^2)\2),sqrtint(n-2*x^2),x^2+x*y+y^2==n&&(!d||!printf("%d",[x,y]))))\\ Set 2nd arg = 1 to print all decompositions, with 0 <= x <= y. - M. F. Hasler, Mar 05 2018

Python

def A088534(n):
    c = 0
    for y in range(n+1):
        if y**2 > n:
            break
        for x in range(y+1):
            z = x*(x+y)+y**2
            if z > n:
                break
            elif z == n:
                c += 1
    return c # Chai Wah Wu, May 16 2022

Formula

a(A003136(n)) > 0; a(A034020(n)) = 0;

a(A118886(n)) > 1; a(A198772(n)) = 1;

a(A198773(n)) = 2; a(A198774(n)) = 3;

a(A198775(n)) = 4;

a(A198799(n)) = n and a(m) <> n for m < A198799(n). - Reinhard Zumkeller, Oct 30 2011, corrected by M. F. Hasler, Mar 05 2018

In the prime factorization of n, let S_1 be the set of distinct prime factors p_i for which p_i == 1 (mod 3), let S_2 be the set of distinct prime factors p_j for which p_j == 2 (mod 3), and let M be the exponent of 3. Then n = 3^M * (Product_{p_i in S_1} p_i ^ e_i) * (Product_{p_j in S_2} p_j ^ e_j), and the number of solutions for x^2 + xy + y^2 = n, 0 <= x <= y is floor((Product_{p_i in S_1} (e_i + 1) + 1) / 2) if all e_j are even and 0 otherwise. E.g. a(1729) = 4 since 1729 = 7^1*13^1*19^1 and floor(((1+1)*(1+1)*(1+1)+1)/2) = 4. - Seth A. Troisi, Jul 02 2020

a(n) = ceiling(A004016(n)/12) = (A002324(n) + A145377(n)) / 2. - Andrey Zabolotskiy, Dec 23 2021

Extensions

Edited by M. F. Hasler, Mar 05 2018