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A316317 Coordination sequence for trivalent node in chamfered version of square grid.

%I A316317 #40 Jun 30 2021 19:54:27
%S A316317 1,3,6,11,14,15,20,25,26,29,34,37,40,43,46,51,54,55,60,65,66,69,74,77,
%T A316317 80,83,86,91,94,95,100,105,106,109,114,117,120,123,126,131,134,135,
%U A316317 140,145,146,149,154,157,160,163,166,171,174,175,180,185,186,189,194
%N A316317 Coordination sequence for trivalent node in chamfered version of square grid.
%H A316317 Rémy Sigrist, <a href="/A316317/b316317.txt">Table of n, a(n) for n = 0..5000</a>
%H A316317 Michel Deza and Mikhail Shtogrin, <a href="https://doi.org/10.1016/S0012-365X(01)00074-7">Isometric embedding of mosaics into cubic lattices</a>, Discrete mathematics 244.1-3 (2002): 43-53. See Fig. 2.
%H A316317 Michel Deza and Mikhail Shtogrin, <a href="/A316316/a316316.png">Isometric embedding of mosaics into cubic lattices</a>, Discrete mathematics 244.1-3 (2002): 43-53. [Annotated scan of page 52 only]
%H A316317 Rémy Sigrist, <a href="/A316317/a316317.gp.txt">PARI program for A316317</a>
%H A316317 Rémy Sigrist, <a href="/A316317/a316317.png">Illustration of first terms</a>
%H A316317 N. J. A. Sloane, <a href="/A316316/a316316_2.png">Initial terms of coordination sequence for trivalent node</a>
%H A316317 N. J. A. Sloane, <a href="/A316316/a316316_7.png">"Basketweave" tiling by 3X1 rectangles which is equivalent (as far as the graph and coordination sequences are concerned) to this tiling </a>
%F A316317 Apparently, a(n + 12) = a(n) + 40 for any n > 0. - _Rémy Sigrist_, Jun 30 2018
%F A316317 This can surely be proved by the Coloring Book Method, although I have not worked out the details. See A316316 for the corresponding proof for a tetravalent node. - _N. J. A. Sloane_, Jun 30 2018
%F A316317 G.f. (assuming above conjecture): (1+x)^2*(1+3*x^2+x^4)/((1-x)^2*(1+x+x^2)*(1+x^2)). - _Robert Israel_, Jul 01 2018
%F A316317 a(n) = (30*n - 9*A056594(n-1) + 6*A102283(n))/9 for n > 0. - Conjectured by _Stefano Spezia_, Jun 12 2021
%o A316317 (PARI) See Links section.
%Y A316317 See A316316 for tetravalent node.
%Y A316317 See A250120 for links to thousands of other coordination sequences.
%Y A316317 Cf. A316358 (partial sums).
%Y A316317 Cf. A056594, A102283.
%K A316317 nonn
%O A316317 0,2
%A A316317 _N. J. A. Sloane_, Jun 29 2018
%E A316317 More terms from _Rémy Sigrist_, Jun 30 2018