A004014 Norms of vectors in the b.c.c. lattice.

0, 3, 4, 8, 11, 12, 16, 19, 20, 24, 27, 32, 35, 36, 40, 43, 44, 48, 51, 52, 56, 59, 64, 67, 68, 72, 75, 76, 80, 83, 84, 88, 91, 96, 99, 100, 104, 107, 108, 115, 116, 120, 123, 128, 131, 132, 136, 139, 140, 144, 147, 148, 152, 155, 160, 163, 164, 168

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Comments

Integers such that A004013(n) is nonzero. - Michael Somos, Jul 28 2014

A subsequence of A047458. The complement seems to be 4*A004215. - Andrey Zabolotskiy, Nov 11 2021

From Mohammed Yaseen, Nov 06 2022: (Start)

These are numbers of the form x^2+y^2+z^2 where x, y and z are either all even (including zero) or all odd.

The selection rule for the planes with Miller indices (hkl) to undergo X-ray diffraction in an f.c.c. lattice is h^2+k^2+l^2 = N where N is a term of this sequence. See A000378 for simple cubic lattice. (End)

References

• J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 116. (Chapter 4 section 6.7)
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Robert Israel, Table of n, a(n) for n = 0..10000
• G. Nebe and N. J. A. Sloane, Home page for this lattice
• Index entries for sequences related to b.c.c. lattice

Crossrefs

Cf. A004013, A047458, A004215.

Union of A034045 and A017101. - Mohammed Yaseen, Nov 06 2022

Programs

Maple

f:= JacobiTheta2(0,z^4)^3+JacobiTheta3(0,z^4)^3:
S:= series(f,z,1001):
select(t -> coeff(S,z,t) <> 0, [$0..1000]); # Robert Israel, Oct 18 2015

Mathematica

f = EllipticTheta[2, 0, z^4]^3 + EllipticTheta[3, 0, z^4]^3; S = f + O[z]^200; Flatten[Position[CoefficientList[S, z], ?Positive] - 1] (* _Jean-François Alcover, Oct 23 2016, after Robert Israel *)

Extensions

More terms from Sean A. Irvine, Oct 17 2015