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A299909 Coordination sequence of node of type 3^6 in 3-uniform tiling #3.54 in the Galebach listing.

%I A299909 #39 Jan 19 2020 18:13:17
%S A299909 1,6,12,18,24,24,30,42,48,48,54,66,66,66,78,90,90,90,102,108,108,114,
%T A299909 126,132,132,138,144,150,156,162,168,174,180,180,186,198,204,204,210,
%U A299909 222,222,222,234,246,246,246,258,264,264,270,282,288,288,294,300,306
%N A299909 Coordination sequence of node of type 3^6 in 3-uniform tiling #3.54 in the Galebach listing.
%C A299909 This tiling has three kinds of nodes. So far the other two types (A299910, A299911) have nor been analyzed.
%H A299909 Colin Barker, <a href="/A299909/b299909.txt">Table of n, a(n) for n = 0..1000</a>
%H A299909 Brian Galebach, <a href="http://probabilitysports.com/tilings.html?u=0&amp;n=3&amp;t=54">Tiling 3.54</a>
%H A299909 Brian Galebach, <a href="/A299909/a299909.png">Tiling 3.54</a> [Annotated figure showing the 3 kinds of points mentioned in A299909, A299910, A299911]
%H A299909 Brian Galebach, <a href="/A250120/a250120.html">k-uniform tilings (k <= 6) and their A-numbers</a>
%H A299909 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">arXiv:1803.08530</a>.
%H A299909 Printable Paper Web Site, <a href="https://www.printablepaper.net/preview/3-3-3-3-3-3_3-3-4-3-4_Tessellation_Paper-Small">Printable 3.3.3.3.3.3,3.3.4.3.4 Tessellation Small</a> [Shows this tiling]
%H A299909 N. J. A. Sloane, <a href="/A299909/a299909_1.png">Labeling of nodes in one sector used to determine the coordination sequence</a>
%H A299909 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (1,-1,1,0,0,0,1,-1,1,-1).
%F A299909 G.f.: (x^10+5*x^9+7*x^8+11*x^7+12*x^6+6*x^5+12*x^4+11*x^3+7*x^2+5*x+1) / ((1-x)^2*(1+x^2)*(x^6+x^5+x^4+x^3+x^2+x+1)).
%F A299909 The denominator can also be written as (1-x)*(1+x^2)*(1-x^7).
%F A299909 Recurrence: (-n^2-5*n)*a(n)-n*a(n+1)+
%F A299909 (-n^2-6*n)*a(n+2)-2*n*a(n+3)-2*n*a(n+4)-2*n*a(n+5)-
%F A299909 2*n*a(n+6)+(n^2+3*n)*a(n+7)-n*a(n+8)+(n^2+4*n)*a(n+9) = 0,
%F A299909 with a(0) = 1, a(1) = 6, a(2) = 12, a(3) = 18, a(4) = 24, a(5) = 24, a(6) = 30, a(7) = 42, a(8) = 48, a(9) = 48.
%F A299909 a(n) = a(n-1) - a(n-2) + a(n-3) + a(n-7) - a(n-8) + a(n-9) - a(n-10) for n>10. - _Colin Barker_, Mar 11 2018
%F A299909 Details of the calculation of the generations function. (Start)
%F A299909 The following lines are written in Maple notation, but should be intelligible as plain text. The colors refer to the labeling of one sector shown in the link.
%F A299909 This analysis did not directly use the "trunks and branches" method described in the Goodman-Strauss & Sloane paper, but was influenced by it.
%F A299909 # The generating function for one of the six sectors:
%F A299909 G:=1+2*x+2*x^2+2*x^3; # green sausages
%F A299909 QG:=G/((1-x^4)*(1-x^7)); # the lattice of green sausages
%F A299909 R:=2+2*x+2*x^2+x^3; # red sausages
%F A299909 QR:=R*(1/(1-x^3))*(x^4/(1-x^4)-x^7/(1-x^7)); # lattice of red sausages
%F A299909 XA:=-x^2/(1-x); # correction for "X-axis"
%F A299909 # red vertical lines of type a
%F A299909 RVLa := x^2/((1-x)*(1-x^4))+x^5*(1/(1-x^3))*(1/(1-x^4)-1/(1-x^7));
%F A299909 # red vertical lines of type b
%F A299909 RVLb:= x^3/((1-x^4)*(1-x^7)) + x^7/((1-x^3)*(1-x^4)) - x^10/((1-x^3)*(1-x^7));
%F A299909 # red vertical lines of type c (twigs to right of vertical sausages)
%F A299909 RVLc:= x^4/((1-x^4)*(1-x^7)) + x^8/((1-x^3)*(1-x^4)) - x^11/((1-x^3)*(1-x^7));
%F A299909 # Total for one sector
%F A299909 T:=QG+QR+XA+RVLa+RVLb+RVLc;
%F A299909 # Grand total, after correcting for overcounting where sectors meet:
%F A299909 U:=6*T-5-6*x;
%F A299909 series(U,x,30);
%F A299909 # After simplification, grand total is:
%F A299909 (x^10+5*x^9+7*x^8+11*x^7+12*x^6+6*x^5+12*x^4+11*x^3+7*x^2+5*x+1) / ((1-x)^2*(1+x^2)*(x^6+x^5+x^4+x^3+x^2+x+1));
%F A299909 (End) (These details added by _N. J. A. Sloane_, Apr 10 2018)
%t A299909 Join[{1}, LinearRecurrence[{1, -1, 1, 0, 0, 0, 1, -1, 1, -1}, {6, 12, 18, 24, 24, 30, 42, 48, 48, 54}, 60]] (* _Jean-François Alcover_, Jan 09 2019 *)
%o A299909 (PARI) Vec((x^10+5*x^9+7*x^8+11*x^7+12*x^6+6*x^5+12*x^4+11*x^3+7*x^2+5*x+1) / ((1-x)^2*(1+x^2)*(x^6+x^5+x^4+x^3+x^2+x+1)) + O(x^60)) \\ _Colin Barker_, Mar 11 2018
%Y A299909 See A299910, A299911 for the other two kinds of nodes.
%K A299909 nonn,easy
%O A299909 0,2
%A A299909 _N. J. A. Sloane_, Mar 07 2018