

A088534


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


14



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
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OFFSET

0,50


REFERENCES

B. C. Berndt, "On a certain thetafunction in a letter of Ramanujan from Fitzroy House", Ganita 43 (1992), 3343.


LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Oscar Marmon, Hexagonal Lattice Points on Circles, arXiv:math/0508201 [math.NT], 2005.


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


EXAMPLE

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)


MATHEMATICA

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


PROG

(PARI) a(n)=sum(i=0, n, sum(j=0, i, if(i^2+i*j+j^2n, 0, 1)))
(PARI) A088534(n, d)=sum(x=0, sqrt(n\3), sum(y=max(x, sqrtint(nx^2)\2), sqrtint(n2*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
(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


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).
Sequence in context: A308264 A065335 A230264 * A178602 A216279 A025441
Adjacent sequences: A088531 A088532 A088533 * A088535 A088536 A088537


KEYWORD

nonn


AUTHOR

Benoit Cloitre, Nov 16 2003


EXTENSIONS

Edited by M. F. Hasler, Mar 05 2018


STATUS

approved



