A299909 Coordination sequence of node of type 3^6 in 3-uniform tiling #3.54 in the Galebach listing.
1, 6, 12, 18, 24, 24, 30, 42, 48, 48, 54, 66, 66, 66, 78, 90, 90, 90, 102, 108, 108, 114, 126, 132, 132, 138, 144, 150, 156, 162, 168, 174, 180, 180, 186, 198, 204, 204, 210, 222, 222, 222, 234, 246, 246, 246, 258, 264, 264, 270, 282, 288, 288, 294, 300, 306
Offset: 0
Links
- Colin Barker, Table of n, a(n) for n = 0..1000
- Brian Galebach, Tiling 3.54
- Brian Galebach, Tiling 3.54 [Annotated figure showing the 3 kinds of points mentioned in A299909, A299910, A299911]
- Brian Galebach, k-uniform tilings (k <= 6) and their A-numbers
- Chaim Goodman-Strauss and N. J. A. Sloane, A Coloring Book Approach to Finding Coordination Sequences, Acta Cryst. A75 (2019), 121-134, also on NJAS's home page. Also arXiv:1803.08530.
- Printable Paper Web Site, Printable 3.3.3.3.3.3,3.3.4.3.4 Tessellation Small [Shows this tiling]
- N. J. A. Sloane, Labeling of nodes in one sector used to determine the coordination sequence
- Index entries for linear recurrences with constant coefficients, signature (1,-1,1,0,0,0,1,-1,1,-1).
Programs
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Mathematica
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 *) -
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
Formula
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)).
The denominator can also be written as (1-x)*(1+x^2)*(1-x^7).
Recurrence: (-n^2-5*n)*a(n)-n*a(n+1)+
(-n^2-6*n)*a(n+2)-2*n*a(n+3)-2*n*a(n+4)-2*n*a(n+5)-
2*n*a(n+6)+(n^2+3*n)*a(n+7)-n*a(n+8)+(n^2+4*n)*a(n+9) = 0,
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.
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
Details of the calculation of the generations function. (Start)
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.
This analysis did not directly use the "trunks and branches" method described in the Goodman-Strauss & Sloane paper, but was influenced by it.
# The generating function for one of the six sectors:
G:=1+2*x+2*x^2+2*x^3; # green sausages
QG:=G/((1-x^4)*(1-x^7)); # the lattice of green sausages
R:=2+2*x+2*x^2+x^3; # red sausages
QR:=R*(1/(1-x^3))*(x^4/(1-x^4)-x^7/(1-x^7)); # lattice of red sausages
XA:=-x^2/(1-x); # correction for "X-axis"
# red vertical lines of type a
RVLa := x^2/((1-x)*(1-x^4))+x^5*(1/(1-x^3))*(1/(1-x^4)-1/(1-x^7));
# red vertical lines of type b
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));
# red vertical lines of type c (twigs to right of vertical sausages)
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));
# Total for one sector
T:=QG+QR+XA+RVLa+RVLb+RVLc;
# Grand total, after correcting for overcounting where sectors meet:
U:=6*T-5-6*x;
series(U,x,30);
# After simplification, grand total is:
(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));
(End) (These details added by N. J. A. Sloane, Apr 10 2018)
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