A397625 Coordination sequence of the bcs net.

1, 6, 24, 56, 98, 152, 218, 296, 386, 488, 602, 728, 866, 1016, 1178, 1352, 1538, 1736, 1946, 2168, 2402, 2648, 2906, 3176, 3458, 3752, 4058, 4376, 4706, 5048, 5402, 5768, 6146, 6536, 6938, 7352, 7778, 8216, 8666, 9128, 9602, 10088, 10586, 11096, 11618

Offset

0,2

Comments

The bcs net is the vertex-transitive (uninodal, in RCSR terminology) 6-coordinated periodic net with RCSR symbol bcs (space group Ia-3d).

Computed by breadth-first search on the periodic graph, cross-checked by an independent breadth-first search on the RCSR Systre-key quotient graph.

For n >= 3, a(n) = A005897(n) = 6*n^2 + 2; the two sequences differ only at n = 1 and n = 2. The bcs net thus shares its asymptotic shell growth with the lattice sequence A005897 while having a different local structure (coordination number 6 rather than 8).

Links

Steven Kotlarz, Table of n, a(n) for n = 0..1000

Reticular Chemistry Structure Resource, The bcs net

Formula

a(0) = 1, a(1) = 6, a(2) = 24; a(n) = 6*n^2 + 2 for n >= 3.

G.f.: (x + 1)*(2*x^4 - 6*x^3 + 7*x^2 + 2*x + 1)/(1 - x)^3.

E.g.f.: exp(x)*(2 + 6*x + 6*x^2) - (1 + x)^2. - Stefano Spezia, Jul 08 2026

Example

The 6-coordinated bcs net has a(1) = 6 nearest neighbors and a(2) = 24 second-shell vertices, after which the shell counts follow 6*n^2 + 2 exactly.

Prog

(PARI) Vec((2*x^5 - 4*x^4 + x^3 + 9*x^2 + 3*x + 1)/(1 - x)^3 + O(x^45))
(Python)
def a(n): return 1 if n == 0 else (6 if n == 1 else (24 if n == 2 else 6*n*n + 2))

Crossrefs

Cf. A005897 (agrees for n >= 3).

Sequence in context: A334701 A274205 A009943 * A028595 A002653 A212904

Adjacent sequences: A397615 A397616 A397619 * A397626 A397627 A397628

Keyword

nonn,easy,new

Author

Steven Kotlarz, Jul 02 2026

Status

approved