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));
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