A198799 -id:A198799 - cp's OEIS frontend

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

Benoit Cloitre, Nov 16 2003

nonn

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

			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)

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

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.

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.

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

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

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 *)

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

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

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

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

Edited by M. F. Hasler, Mar 05 2018

A230655 Squared radii of circles around a point of the hexagonal lattice that contain a record number of lattice points.

Original entry on oeis.org

0, 1, 7, 49, 91, 637, 1729, 8281, 12103, 53599, 157339, 375193, 1983163, 4877509, 13882141, 85276009, 180467833, 596932063, 3428888827, 4178524441, 7760116819, 29249671087, 36412855843, 147442219561, 254889990901, 473367125959, 1784229936307, 2439661341481

Offset: 1

Hugo Pfoertner, Oct 27 2013

nonn

It appears that this is also the sequence of numbers with a record number of divisors all of whose prime factors are of the form 3k + 1. - Amiram Eldar, Sep 12 2019 [This is correct, see A343771. - Jianing Song, May 19 2021]

Indices of records of A004016. Apart from the first term, also indices of records of A002324. - Jianing Song, May 20 2021

			a(2)=7 because a circle with radius sqrt(7) around the lattice point at (0,0) is the first circle that passes through more lattice points than a circle with radius 1, which passes through 6 points. The 12 hit points are (+-1/2,+-3*sqrt(3)/2), (+-2,+-sqrt(3)), (+-5/2, +-sqrt(3)/2).

Jianing Song, Table of n, a(n) for n = 1..247 (all terms <= 10^75.)

Cf. A004016, A002324.
Cf. A003136 (all occurring squared radii), A198799 (common terms), A230656 (index positions of records), A344472 (records).
Apart from the first term, subsequence of A343771.
Indices of records of Sum_{d|n} kronecker(m, d): this sequence (m=-3), A071383 (m=-4, similar sequence for square lattice), A279541 (m=-6).

my(v=list_A344473(10^15), rec=0); print1(0, ", "); for(n=1, #v, if(numdiv(v[n])>rec, rec=numdiv(v[n]); print1(v[n], ", "))) \\ Jianing Song, May 20 2021, see program for A344473

Offset corrected by Jianing Song, May 20 2021

A300419 Smallest nonnegative number k such that k can be written in exactly n ways as x^2 + xy + y^2 where x and y are positive integers, with x >= y.

Original entry on oeis.org

0, 3, 91, 637, 1729, 24843, 12103, 405769, 53599, 157339, 593047, 59648043, 375193, 2989441, 8968323, 7709611, 1983163, 3360173089, 4877509, 2339177536969, 18384457, 377770939, 146482609, 439447827, 13882141, 1302924259

Offset: 0

Altug Alkan, M. F. Hasler and Robert G. Wilson, Mar 05 2018

Except a(0) and a(1), all terms are in A118886.
First positive square term of this sequence is a(7) = 405769 = a(3)^2.
a(3), a(7) = a(3)^2 and a(13) = a(4)^2 are also the sum of two nonzero squares in exactly one way.
a(18) = 4877509, a(20) = 18384457, a(22) = 146482609, a(24) = 13882141, a(27) = 92672671, a(30) = 238997941, a(32) = 85276009, a(36) = 180467833. - Robert G. Wilson v, Mar 06 2018

			a(2) = 91 because 91 = 1^2 + 1*9 + 9^2 = 5^2 + 5*6 + 6^2 and 91 is the least number with this property.

Robert G. Wilson v, Solutions of a(n) for n <= 16

Cf. A024614, A118886, A198799, A327796, A327797.

nmx = 4750; t = Split@ Sort@ Flatten@ Table[x^2 + x*y + y^2, {x, nmx}, {y, x, nmx}]; lmt = 1 + Length@ t; f[n_] := Block[{k = 1}, While[Length@ t[[k]] != n && k < lmt, k++]; t[[k]][[1]]]; Array[f, 16] (* Robert G. Wilson v, Mar 06 2018 *)

N(n,d)=sum(x=1,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 display all decompositions.
a(n)=for(k=0,oo,N(k)==n&&return(k))

If A198799(n) is not a square and there is no square s < A198799(n) such that A088534(s) = n + 1, then a(n) = A198799(n), for all n > 0.

If A198799(n+1) is a square, then a(n) <= A198799(n+1).

a(17)-a(18) from Giovanni Resta, Mar 16 2018

a(19)-a(25) from Bert Dobbelaere, Feb 18 2023

A374275 Smallest k such that k can be written in exactly n ways as x^2 + 3xy + y^2 with 0 <= x <= y.

Original entry on oeis.org

2, 0, 121, 2299, 6061, 43681, 66671, 33659659, 187891, 1266749, 8067191, 639533521, 2066801

Offset: 0

Seiichi Manyama, Jul 02 2024

a(n) is the smallest nonnegative k such that A374276(k) = n.
a(14) = 36735721.
a(15) = 153276629.
a(16) = 7703531.
a(18) = 39269219.
a(20) = 250082921.
a(24) = 84738841.
a(32) = 454508329.

Cf. A000446, A198799.

Cf. A031363, A374091, A374095, A374276.

A307109 Number of integer solutions to x^2 + x*y + y^2 <= n excluding (0,0), divided by 2.

Original entry on oeis.org

1, 1, 4, 5, 5, 5, 11, 11, 12, 12, 12, 15, 21, 21, 21, 22, 22, 22, 28, 28, 34, 34, 34, 34, 35, 35, 38, 44, 44, 44, 50, 50, 50, 50, 50, 51, 57, 57, 63, 63, 63, 63, 69, 69, 69, 69, 69, 72, 79, 79, 79, 85, 85, 85, 85, 85, 91, 91, 91, 91, 97, 97, 103, 104, 104, 104

Offset: 1

Hugo Pfoertner, Mar 25 2019

nonn

			a(1)=a(2)=(2/2)=1: 2 solutions (-1,1),(1,-1);
a(3)=(8/2)=4: 2 solutions for n<3 and the 6 solutions (1,1), (-1,-1), (1,-2), (2,-1), (-1,2), (-2,1);
a(4)=a(5)=a(6)=10/2=5: 8 solutions for n<6 and the 2 solutions (-2,2), (2,-2).

Cf. A014201, A014202, A198799, A230655, A300419.

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

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A230655 Squared radii of circles around a point of the hexagonal lattice that contain a record number of lattice points.

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A300419 Smallest nonnegative number k such that k can be written in exactly n ways as x^2 + xy + y^2 where x and y are positive integers, with x >= y.

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A374275 Smallest k such that k can be written in exactly n ways as x^2 + 3*x*y + y^2 with 0 <= x <= y.

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A307109 Number of integer solutions to x^2 + x*y + y^2 <= n excluding (0,0), divided by 2.

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A374275 Smallest k such that k can be written in exactly n ways as x^2 + 3xy + y^2 with 0 <= x <= y.