OFFSET
0,2
COMMENTS
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]
Index entries for linear recurrences with constant coefficients, signature (1,-1,1,0,0,0,1,-1,1,-1).
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)
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Mar 07 2018
STATUS
approved