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A003136
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Loeschian numbers: numbers of the form x^2 + xy + y^2; norms of vectors in A2 lattice.
(Formerly M2336)
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80
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0, 1, 3, 4, 7, 9, 12, 13, 16, 19, 21, 25, 27, 28, 31, 36, 37, 39, 43, 48, 49, 52, 57, 61, 63, 64, 67, 73, 75, 76, 79, 81, 84, 91, 93, 97, 100, 103, 108, 109, 111, 112, 117, 121, 124, 127, 129, 133, 139, 144, 147, 148, 151, 156, 157, 163, 169, 171, 172, 175, 181, 183, 189, 192
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OFFSET
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1,3
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COMMENTS
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Equally, numbers of the form x^2 - xy + y^2. - Ray Chandler, Jan 27 2009
Also, numbers of the form X^2+3Y^2 (X=y+x/2, Y=x/2), cf. A092572. - Zak Seidov, Jan 20 2009
Theorem (Schering, Delone, Watson): The only positive definite binary quadratic forms that represent the same numbers are x^2+xy+y^2 and x^2+3y^2 (up to scaling). - N. J. A. Sloane, Jun 22 2014
Equivalently, numbers n such that the coefficient of x^n in Theta3(x)*Theta3(x^3) is nonzero. - Joerg Arndt, Jan 16 2011
Equivalently, numbers n such that the coefficient of x^n in a(x) (resp. b(x)) is nonzero where a(), b() are cubic AGM functions. - Michael Somos, Jan 16 2011
Relative areas of equilateral triangles whose vertices are on a triangular lattice. - Anton Sherwood (bronto(AT)pobox.com), Apr 05 2001
2 appended to a(n) (for positive n) corresponds to capsomere count in viral architectural structures (cf. A071336). - Lekraj Beedassy, Apr 14 2006
The triangle in A132111 gives the enumeration: n^2 + k*n + k^2, 0<=k<=n.
The number of coat proteins at each corner of a triangular face of a virus shell. - Parthasarathy Nambi, Sep 04 2007
A088534(a(n)) > 0. - Reinhard Zumkeller, Oct 30 2011
Numbers of the form (x^2 + y^2 + (x + y)^2)/2. If we let z = - x - y, then all the solutions to x^2 + y^2 + z^2 = k with x + y + z = 0 are k = 2a(n) for any n. - Jon Perry, Dec 16 2012
Sequence of divisors of the hexagonal lattice, except zero (where it is said that an integer n divides a lattice if there exists a sublattice of index n; example: 3 divides the hexagonal lattice). - Jean-Christophe Hervé, May 01 2013
Numbers of the form - (x*y + y*z + x*z) with x + y + z = 0. Numbers of the form x^2 + y^2 + z^2 - (x*y + y*z + x*z) = (x - y)*(x - z) + (y - x) * (y - z) + (z - x) * (z - y). - Michael Somos, Jun 26 2013
Equivalently, the existence spectrum of affine Mendelsohn triple systems, cf. A248107. - David Stanovsky, Nov 25 2014
Lame's solutions to the Helmholtz equation with Dirichlet boundary conditions on the unit-edged equilateral triangle have eigenvalues of the form: (x^2+x*y+y^2)*(4*Pi/3)^2. The actual set, starting at 1 and counting degeneracies, is given by A060428, e.g., the first degeneracy is 49 where (x,y)=(0,7) and (3,5). - Robert Stephen Jones, Oct 01 2015
The number of structure units in an icosahedral virus is 20*a(n) for an integer n, see Stannard link. - Charles R Greathouse IV, Nov 03 2015
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REFERENCES
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J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 111.
B. N. Delone. Geometry of positive quadratic forms, addendum, Uspechi Mat. Sri, 4 (1938), 102-164.
J. U. Marshall, The Loeschian numbers as a problem in number theory, Geographical Analysis, 7 (1975), 421-426.
J. U. Marshall, The construction of the Loschian landscape, Geographical Analysis, 9 (Jan. 1977), 1-13.
J. U. Marshall, Christallerian networks in the Loschian economic landscape, Professional Geographer, 29 (No. 2, 1977), 153-159.
Ivars Peterson, "The Jungles of Randomness: A Mathematical Safari", John Wiley and Sons, (1998) pp. 53.
M. Schering. Theoremes relatifs aux formes quadratiques qui representent les memes nombres, J. Math, pures el appl., 2 serie 4 (1859).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
V. N. Timofeev. "On positive quadratic forms, representing the same numbers", Uspekhi Mat. Nauk, 18 (1963), 191-193.
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LINKS
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T. D. Noe, Table of n, a(n) for n = 1..10000
M. Bernstein, N. J. A. Sloane and P. E. Wright, On Sublattices of the Hexagonal Lattice, Discrete Math. 170 (1997) 29-39 (Abstract, pdf, ps).
J. H. Conway, E. M. Rains and N. J. A. Sloane, On the existence of similar sublattices, Canad. J. Math. 51 (1999), 1300-1306 (Abstract, pdf, ps).
Diane M. Donovan, Terry S. Griggs, Thomas A. McCourt, Jakub Opršal, David Stanovský, Distributive and anti-distributive Mendelsohn triple systems, arXiv:1411.5194 [math.CO], (19-November-2014)
William C. Jagy and Irving Kaplansky, Positive definite binary quadratic forms that represent the same primes [Cached copy]
Y. Kitaoka, On the relation between the positive-definite quadratic forms with the same representation numbers, Proc. Jap. Acad., 47 (1971), 439-441.
August Lösch, Economics of Location (1954), see pp. 117f.
U. P. Nair, Elementary results on the binary quadratic form a^2+ab+b^2, arXiv:math/0408107 [math.NT], 2004.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
Linda Stannard, Principles of Virus Architecture (archived version)
G. L. Watson, Determination of a binary quadratic form by its values at integer points, Mathematika 26 (1979), no. 1, 72--75. MR0557128 (81e:10019).
G. L. Watson, Acknowledgement: Determination of a binary quadratic form by its values at integer points, Mathematika 27 (1980), no. 2, 188 (1981). MR0610704 (82d:10037)
Index entries for sequences related to A2 = hexagonal = triangular lattice
Index entries for "core" sequences
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FORMULA
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Either n=0 or else in the prime factorization of n all primes of the form 3a+2 must occur to even powers only (there is no restriction of primes congruent to 0 or 1 mod 3).
If n is in the sequence, then n^k is in the sequence (but the converse is not true). n is in the sequence iff n^(2k+1) is in the sequence. - Ray Chandler, Feb 03 2009
The sequence is multiplicative in the sense that if m and n are in the sequence, so is m*n. - Jon Perry, Dec 18 2012
Comments from _Richard Schroeppel_, Jul 20 2016 (Start):
The set is also closed under restricted division: If M and N are members, and M divides N, then N/M is a member.
If N == 2 (mod3), N is not in the sequence.
The density of members (relative to the integers>0) gradually falls to 0. The density goes as O(1/sqrt(log N)). This implies that, if N is a member, the average expected number of representations of N is O(sqrt(log N)).
Representations usually come in sets of 6: (K,L), (K+L,-K), (K+L,-L) and their negatives. (End)
Comment from Victor S. Miller, Jul 20 2016: Since Q(zeta), where zeta is a primitive 3rd root of unity has class number 1, the situation as to whether an integer is of the form x^2 + xy + y^2 is similar to the situation with x^2 + y^2: n is of the that form if and only if every prime p dividing n which is = 5 mod 6 divides it to an even power. The density of of 1/sqrt(x) that Rich mentioned is an old result due to Landau.
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MAPLE
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readlib(ifactors): for n from 2 to 200 do m := ifactors(n)[2]: flag := 1: for i from 1 to nops(m) do if m[i, 1] mod 3 = 2 and m[i, 2] mod 2 = 1 then flag := 0; break fi: od: if flag=1 then printf(`%d, `, n) fi: od: # James A. Sellers, Dec 07 2000
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MATHEMATICA
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ok[n_] := Resolve[Exists[{x, y}, Reduce[n == x^2 + x*y + y^2, {x, y}, Integers]]]; Select[Range[0, 192], ok] (* Jean-François Alcover, Apr 18 2011 *)
nn = 14; Select[Union[Flatten[Table[x^2 + x*y + y^2, {x, 0, nn}, {y, 0, x}]]], # <= nn^2 &] (* T. D. Noe, Apr 18 2011 *)
QP = QPochhammer; s = QP[q]^3 / QP[q^3]/3 + O[q]^200; Position[ CoefficientList[s, q], n_ /; n != 0] - 1 // Flatten (* Jean-François Alcover, Nov 27 2015, adapted from PARI *)
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PROG
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(Haskell)
import Data.Set (singleton, union, fromList, deleteFindMin)
a003136 n = a003136_list !! (n-1)
a003136_list = f 0 $ singleton 0 where
f x s | m < x ^ 2 = m : f x s'
| otherwise = m : f x'
(union s' $ fromList $ map (\y -> x'^2+(x'+y)*y) [0..x'])
where x' = x + 1
(m, s') = deleteFindMin s
-- Reinhard Zumkeller, Oct 30 2011
(PARI) isA003136(n)=local(fac, flag); if(n==0, 1, fac=factor(n); flag=1; for(i=1, matsize(fac)[1], if(Mod(fac[i, 1], 3)==2 && Mod(fac[i, 2], 2)==1, flag=0)); flag)
(PARI) is(n)=#bnfisintnorm(bnfinit(z^2+z+1), n) \\ Ralf Stephan, Oct 18 2013
(PARI) x='x+O('x^200); p=eta(x)^3/eta(x^3); for(n=0, 199, if(polcoeff(p, n) != 0, print1(n, ", "))) \\ Altug Alkan, Nov 08 2015
(MAGMA) [n: n in [0..192] | NormEquation(3, n) eq true]; // Arkadiusz Wesolowski, May 11 2016
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CROSSREFS
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Subsequence of A032766.
Cf. A004611, A034017, A045897, A060428.
See A092572 for numbers of form x^2 + 3 y^2 with positive x,y.
Cf. A034020 (complement), A007645 (primes); partitions: A198726, A198727; subsequences: A198772, A118886, A198773, A198774, A198775.
Subsequences: A202822, A260682.
Sequence in context: A120451 A060428 A035238 * A034022 A198772 A185256
Adjacent sequences: A003133 A003134 A003135 * A003137 A003138 A003139
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KEYWORD
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core,easy,nonn,nice
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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