A397625 - OEIS

%I #18 Jul 13 2026 02:32:24

%S 1,6,24,56,98,152,218,296,386,488,602,728,866,1016,1178,1352,1538,

%T 1736,1946,2168,2402,2648,2906,3176,3458,3752,4058,4376,4706,5048,

%U 5402,5768,6146,6536,6938,7352,7778,8216,8666,9128,9602,10088,10586,11096,11618

%N Coordination sequence of the bcs net.

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

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

%C 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).

%H Steven Kotlarz, <a href="/A397625/b397625.txt">Table of n, a(n) for n = 0..1000</a>

%H Reticular Chemistry Structure Resource, <a href="http://rcsr.net/nets/bcs">The bcs net</a>

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

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

%F E.g.f.: exp(x)*(2 + 6*x + 6*x^2) - (1 + x)^2. - _Stefano Spezia_, Jul 08 2026

%e 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.

%o (PARI) Vec((2*x^5 - 4*x^4 + x^3 + 9*x^2 + 3*x + 1)/(1 - x)^3 + O(x^45))

%o (Python)

%o def a(n): return 1 if n == 0 else (6 if n == 1 else (24 if n == 2 else 6*n*n + 2))

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

%K nonn,easy,new

%O 0,2

%A _Steven Kotlarz_, Jul 02 2026