1,3

Equally, numbers of form x^2 - xy + y^2. [From Ray Chandler (rayjchandler(AT)sbcglobal.net), Jan 27 2009]

Also, numbers of form X^2+3Y^2 (X=y+x/2, Y=x/2). Cf. A092572 Numbers of the form x^2+3y^2 where x and y are positive integers. [From Zak Seidov (zakseidov(AT)yahoo.com), Jan 20 2009].

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 (blekraj(AT)yahoo.com), 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 (PachaNambi(AT)yahoo.com), Sep 04 2007

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 111.

J. U. Marshall, The Loeschian numbers as a problem in number theory, Geographical Analysis, 7 (1975), 421-426.

Ivars Peterson, "The Jungles of Randomness: A Mathematical Safari", John Wiley and Sons, (1998) pp. 53.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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).

U. P. Nair, Elementary results on the binary quadratic form a^2+ab+b^2

Index entries for sequences related to A2 = hexagonal = triangular lattice

Index entries for "core" sequences

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. [From Ray Chandler (rayjchandler(AT)sbcglobal.net), Feb 03 2009]

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: # from James A. Sellers Dec 07 2000

Cf. A004611, A034017, A045897, A060428.

See A092572 for numbers of form x^2 + 3 y^2 with positive x,y.

Sequence in context: A120451 A060428 A035238 this_sequence A034022 A070992 A060142

Adjacent sequences: A003133 A003134 A003135 this_sequence A003137 A003138 A003139

core,easy,nonn,nice

N. J. A. Sloane (njas(AT)research.att.com).