This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
Divides naturally into blocks of sizes 1,2,4,8,16,...: 1, 5, 9, 21, 25, 37, 57, 85, 89, 101, 121, 149, 169, 213, 281, <- terms 8 through 15 341, 345, 357, 377, 405, 425, 469, 537, 597, 617, 661, 729, 805, 889, 1045, 1241, 1365, 1369, 1381, 1401, 1429, 1449, 1493, 1561, 1621, 1641, 1685, 1753, 1829, 1913, 2069, 2265, 2389, 2409, 2453, 2521, ... From _Omar E. Pol_, Feb 18 2015: (Start) Also, written as an irregular triangle T(j,k), k>=1, in which the row lengths are the terms of A011782: 1; 5; 9, 21; 25, 37, 57, 85; 89, 101, 121, 149, 169, 213, 281, 341; 345, 357, 377, 405, 425, 469, 537, 597, 617, 661, 729, 805, 889, 1045, 1241, 1365; The right border gives the positive terms of A002450. It appears that T(j,k) = A162795(j,k) = A147562(j,k), if k is a power of 2, for example: it appears that the three mentioned triangles only share the elements from the columns 1, 2, 4, 8, 16, ... (End)
(Maple program that uses the actual definition of the automaton, rather than the (conjectured) formula, from N. J. A. Sloane, Feb 15 2015): # Count terms in a polynomial: C := f->`if`(type(f, `+`), nops(f), 1); # Replace all nonzero coeffts by 1: bool := proc(f) local ix, iy, f2, i, t1, t2, A; f2:=expand(f); if whattype(f) = `+` then t1:=nops(f2); A:=0; for i from 1 to t1 do t2:=op(i, f2); ix:=degree(t2, x); iy:=degree(t2, y); A:=A+x^ix*y^iy; od: A; else ix:=degree(f2, x); iy:=degree(f2, y); x^ix*y^iy; fi; end; # a loop that produces M steps of A169707 and A169708: M:=20; F:=x*y+x/y+1/x*y+1/x/y mod 2; GG[0]:=1; for n from 1 to M do dd[n]:=expand(F*GG[n-1]) mod 2; GG[n]:=bool(GG[n-1]+dd[n]); lprint(n,C(GG[n]), C(GG[n]-GG[n-1])); od:
Map[Function[Apply[Plus,Flatten[ #1]]], CellularAutomaton[{ 750, {2,{{0,2,0},{2,1,2},{0,2,0}}},{1,1}},{{{1}},0},100]]
ArrayPlot /@ CellularAutomaton[{750, {2, {{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}}, {1, 1}}, {{{1}}, 0}, 23]
(* The next two lines deal with the equivalent CA based on neighbors NW, NE, SE, SW. This is to facilitate the comparison with A246333 and A246335 *)
Map[Function[Apply[Plus, Flatten[ #1]]], CellularAutomaton[{ 750, {2, {{2, 0, 2}, {0, 1, 0}, {2, 0, 2}}}, {1, 1}}, {{{1}}, 0}, 100]]
ArrayPlot /@ CellularAutomaton[{750, {2, {{2, 0, 2}, {0, 1, 0}, {2, 0, 2}}}, {1, 1}}, {{{1}}, 0}, 23]
a(43) = (1705 + 1753)/2 = 3458/2 = 1729.
CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
code=1006; stages=128;
rule=IntegerDigits[code,2,10];
g=2*stages+1; (* Maximum size of grid *)
a=PadLeft[{{1}},{g,g},0,Floor[{g,g}/2]]; (* Initial ON cell on grid *)
ca=a;
ca=Table[ca=CAStep[rule,ca],{n,1,stages+1}];
PrependTo[ca,a];
(* Trim full grid to reflect growth by one cell at each stage *)
k=(Length[ca[[1]]]+1)/2;
ca=Table[Table[Part[ca[[n]][[j]],Range[k+1-n,k-1+n]],{j,k+1-n,k-1+n}],{n,1,k}];
on=Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)
Table[Total[Part[on,Range[1,i]]],{i,1,Length[on]}] (* Sum at each stage *)
CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
code=1006; stages=128;
rule=IntegerDigits[code,2,10];
g=2*stages+1; (* Maximum size of grid *)
a=PadLeft[{{1}},{g,g},0,Floor[{g,g}/2]]; (* Initial ON cell on grid *)
ca=a;
ca=Table[ca=CAStep[rule,ca],{n,1,stages+1}];
PrependTo[ca,a];
(* Trim full grid to reflect growth by one cell at each stage *)
k=(Length[ca[[1]]]+1)/2;
ca=Table[Table[Part[ca[[n]][[j]],Range[k+1-n,k-1+n]],{j,k+1-n,k-1+n}],{n,1,k}];
on=Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)
Part[on,2^Range[0,Log[2,stages]]] (* Extract relevant terms *)
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