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Search: id:A003136
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| A003136 |
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Loeschian numbers: of form x^2 + xy + y^2; norms of vectors in A2 lattice.
(Formerly M2336)
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+0
21
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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
(list; graph; listen)
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OFFSET
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1,3
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COMMENT
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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
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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.
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.
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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).
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
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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).
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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: # from James A. Sellers Dec 07 2000
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CROSSREFS
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Cf. A004611, A034017, A045897, A060428.
Essentially the same as A035328.
Sequence in context: A120451 A060428 A035238 this_sequence A034022 A070992 A060142
Adjacent sequences: A003133 A003134 A003135 this_sequence 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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njas
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