A255292
Number of 2's in expansion of F^n mod 3, where F = 1/x+1+x+1/y+y.
Original entry on oeis.org
0, 0, 9, 0, 0, 37, 9, 45, 44, 0, 0, 45, 0, 0, 177, 37, 185, 156, 9, 45, 72, 45, 225, 228, 44, 220, 573, 0, 0, 45, 0, 0, 185, 45, 225, 220, 0, 0, 225, 0, 0, 877, 177, 885, 716, 37, 185, 256, 185, 925, 788, 156, 780, 2281, 9
Offset: 0
The pairs [no. of 1's, no. of 2's] are [1, 0], [5, 0], [4, 9], [5, 0], [25, 0], [12, 37], [4, 9], [20, 45], [69, 44], [5, 0], [25, 0], [20, 45], [25, 0], [125, 0], [52, 177], [12, 37], [60, 185], [281, 156], [4, 9], [20, 45], [97, 72], [20, 45], [100, 225], [353, 228], [69, 44], [345, 220], [448, 573], [5, 0], [25, 0], [20, 45], ...
-
# C3 Counts 1's and 2's
C3 := proc(f) local c,ix,iy,f2,i,t1,t2,n1,n2;
f2:=expand(f) mod 3; n1:=0; n2:=0;
if whattype(f2) = `+` then
t1:=nops(f2);
for i from 1 to t1 do t2:=op(i, f2); ix:=degree(t2, x); iy:=degree(t2, y);
c:=coeff(coeff(t2,x,ix),y,iy);
if (c mod 3) = 1 then n1:=n1+1; else n2:=n2+1; fi; od: RETURN([n1,n2]);
else ix:=degree(f2, x); iy:=degree(f2, y);
c:=coeff(coeff(f2,x,ix),y,iy);
if (c mod 3) = 1 then n1:=n1+1; else n2:=n2+1; fi; RETURN([n1,n2]);
fi;
end;
F3:=1/x+1+x+1/y+y mod 3;
g:=(F,n)->expand(F^n) mod 3;
[seq(C3(g(F3,n))[2],n=0..60)];
A255293
Number of 1's in expansion of F^n mod 3, where F = 1/x+2+x+1/y+y.
Original entry on oeis.org
1, 4, 8, 4, 17, 29, 8, 37, 49, 4, 17, 37, 17, 76, 128, 29, 136, 196, 8, 37, 89, 37, 176, 292, 49, 260, 584, 4, 17, 37, 17, 76, 136, 37, 176, 260, 17, 76, 176, 76, 353, 605, 128, 613, 961, 29, 136, 332, 136, 653, 1105, 196
Offset: 0
The pairs [no. of 1's, no. of 2's] are [1, 0], [4, 1], [8, 5], [4, 1], [17, 8], [29, 20], [8, 5], [37, 28], [49, 64], [4, 1], [17, 8], [37, 28], [17, 8], [76, 49], [128, 101], [29, 20], [136, 109], [196, 241], [8, 5], [37, 28], [89, 80], [37, 28], [176, 149], [292, 289], [49, 64], [260, 305], [584, 437], [4, 1], [17, 8], [37, 28], ...
-
# C3 Counts 1's and 2's
C3 := proc(f) local c,ix,iy,f2,i,t1,t2,n1,n2;
f2:=expand(f) mod 3; n1:=0; n2:=0;
if whattype(f2) = `+` then
t1:=nops(f2);
for i from 1 to t1 do t2:=op(i, f2); ix:=degree(t2, x); iy:=degree(t2, y);
c:=coeff(coeff(t2,x,ix),y,iy);
if (c mod 3) = 1 then n1:=n1+1; else n2:=n2+1; fi; od: RETURN([n1,n2]);
else ix:=degree(f2, x); iy:=degree(f2, y);
c:=coeff(coeff(f2,x,ix),y,iy);
if (c mod 3) = 1 then n1:=n1+1; else n2:=n2+1; fi; RETURN([n1,n2]);
fi;
end;
F4:=1/x+2+x+1/y+y mod 3;
g:=(F,n)->expand(F^n) mod 3;
[seq(C3(g(F4,n))[1],n=0..60)];
A138276
Total number of active nodes of the Rule 150 cellular automaton on an infinite Bethe lattice with coordination number 3 (with a single 1 as initial condition).
Original entry on oeis.org
1, 4, 6, 18, 30, 90, 102, 306, 510, 1530, 1542, 4626, 7110
Offset: 0
Jens Christian Claussen (claussen(AT)theo-physik.uni-kiel.de), Mar 11 2008
Let x_0 be the state (0 or 1) of the focal node and x_i the state of every node that is i steps away from the focal node. In time step n=0, all x_i=0 except x_0=1 (start with a single seed). In the next step, x_1=1 as they have 1 neighbor being 1. For n=2, the x_1 nodes have 1 neighbor being 1 (x_0) and
themselves being 1; the sum being 2, modulo 2, resulting in x_1=0. The focal node itself is 1 and has 3 neighbors being 1, sum being 4, modulo 2, resulting in x_0=0. The outmost nodes x_n are always 1.
Thus one has the patterns
x_0, x_1, x_2, ...
1
1 1
0 0 1
0 0 1 1
0 0 1 0 1
0 0 1 1 1 1
0 0 1 0 0 0 1
0 0 1 1 0 0 1 1
0 0 1 0 1 0 1 0 1
0 0 1 1 1 1 1 1 1 1
0 0 1 0 0 0 0 0 0 0 1
After 2 time steps, the x_0 and x_1 stay frozen at zero and the remaining x_i are generated by Rule 60 (or Rule 90 on half lattice spacing).
These nodes have multiplicities 1,3,6,12,24,48,96,192,384,768,...
The sequence then is obtained by
a(n) = x_0(n) + 3 * Sum_{i=1..n} x_i(n) * 2^(i-1)
A323110
Aggregate values of n-th stage of growth for two-dimensional cellular automaton defined by "Rule 614", based on the 5-celled von Neumann neighborhood, calculated via even-zeroing instead of mod 2.
Original entry on oeis.org
1, 5, 9, 21, 9, 45, 57, 213, 129, 645, 1161, 2709, 657, 3285, 5841, 13869, 3129, 15645, 28161, 65709, 28161, 140805, 178353, 666477, 391137, 1955685, 3520233, 8213877, 3518721, 17593605, 22290273, 83271357, 48828129, 244140645, 439453161, 1025390709, 439453161, 2197265805, 2783203353, 10400391477
Offset: 0
Comments