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%I #74 Jun 30 2021 19:52:07
%S 1,4,8,8,12,20,20,20,28,32,32,36,40,44,48,48,52,60,60,60,68,72,72,76,
%T 80,84,88,88,92,100,100,100,108,112,112,116,120,124,128,128,132,140,
%U 140,140,148,152,152,156,160,164,168,168,172,180,180,180,188,192,192
%N Coordination sequence for tetravalent node in chamfered version of square grid.
%H Rémy Sigrist, <a href="/A316316/b316316.txt">Table of n, a(n) for n = 0..5000</a>
%H 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 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 Michel Deza and Mikhail Shtogrin, <a href="/A316316/a316316_5.png">Enlargement of figure from previous link</a>
%H Chaim Goodman-Strauss and N. J. A. Sloane, <a href="https://doi.org/10.1107/S2053273318014481">A Coloring Book Approach to Finding Coordination Sequences</a>, Acta Cryst. A75 (2019), 121-134, also <a href="http://NeilSloane.com/doc/Cairo_final.pdf">on NJAS's home page</a>. Also <a href="http://arxiv.org/abs/1803.08530">on arXiv</a>, arXiv:1803.08530 [math.CO], 2018-2019.
%H Rémy Sigrist, <a href="/A316316/a316316.gp.txt">PARI program for A316316</a>
%H Rémy Sigrist, <a href="/A316316/a316316_6.png">Illustration of first terms</a>
%H N. J. A. Sloane, <a href="/A316316/a316316_1.png">Initial terms of coordination sequence for tetravalent node</a>
%H N. J. A. Sloane, <a href="/A316316/a316316_3.png">Trunks and branches structure of tetravalent node</a> (First part of proof that a(n+12)=a(n)+40).
%H N. J. A. Sloane, <a href="/A316316/a316316_4.png">Calculation of coordination sequence</a> (Second part of proof that a(n+12)=a(n)+40).
%H 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>
%H N. J. A. Sloane, <a href="/A316316/a316316.pdf">An equivalent tiling seen on the sidewalk of East 70th St in New York City</a>. As far as the graph and coordination sequences are concerned, this is the same as the chamfered square grid. The trivalent vertices labeled b and c are equivalent to each other.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,-1,2,-1,1,-1).
%F Apparently, a(n + 12) = a(n) + 40 for any n > 0. - _Rémy Sigrist_, Jun 30 2018
%F From _N. J. A. Sloane_, Jun 30 2018: This conjecture is true.
%F Theorem: a(n + 12) = a(n) + 40 for any n > 0.
%F The proof uses the Coloring Book Method described in the Goodman-Strauss - Sloane article. For details see the two links.
%F From _Colin Barker_, Dec 13 2018: (Start)
%F G.f.: (1 + 3*x + 5*x^2 + 2*x^3 + 5*x^4 + 3*x^5 + x^6) / ((1 - x)^2*(1 + x^2)*(1 + x + x^2)).
%F a(n) = a(n-1) - a(n-2) + 2*a(n-3) - a(n-4) + a(n-5) - a(n-6) for n>6.
%F (End)
%F a(n) = (2/9)*(15*n + 9*A056594(n-1) - 6*A102283(n)) for n > 0. - _Stefano Spezia_, Jun 12 2021
%t Join[{1}, LinearRecurrence[{1, -1, 2, -1, 1, -1}, {4, 8, 8, 12, 20, 20}, 100]] (* _Jean-François Alcover_, Dec 13 2018 *)
%o (PARI) See Links section.
%Y See A316317 for trivalent node.
%Y See A250120 for links to thousands of other coordination sequences.
%Y Cf. A316357 (partial sums).
%Y Cf. A056594, A102283.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Jun 29 2018
%E More terms from _Rémy Sigrist_, Jun 30 2018
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