A170903 -id:A170903 - cp's OEIS frontend

A160552 a(0)=0, a(1)=1; a(2^i+j) = 2*a(j) + a(j+1) for 0 <= j < 2^i.

0, 1, 1, 3, 1, 3, 5, 7, 1, 3, 5, 7, 5, 11, 17, 15, 1, 3, 5, 7, 5, 11, 17, 15, 5, 11, 17, 19, 21, 39, 49, 31, 1, 3, 5, 7, 5, 11, 17, 15, 5, 11, 17, 19, 21, 39, 49, 31, 5, 11, 17, 19, 21, 39, 49, 35, 21, 39, 53, 59, 81, 127, 129, 63, 1, 3, 5, 7, 5, 11, 17, 15, 5, 11, 17, 19, 21, 39, 49, 31

Offset: 0

David Applegate, May 18 2009

nonn

This recurrence is patterned after the one for A152980, but without the special cases.
Sequence viewed as triangle:
  0,
  1,
  1, 3,
  1, 3, 5, 7,
  1, 3, 5, 7, 5, 11, 17, 15,
  1, 3, 5, 7, 5, 11, 17, 15, 5, 11, 17, 19, 21, 39, 49, 31.
The rows converge to A151548.
Also the sum of the terms in the k-th row (k >= 1) is 4^(k-1). Proof by induction. - N. J. A. Sloane, Jan 21 2010
If this sequence [1, 1, 3, 1, 3, 5, 7, 1, 3, 5, 7, 5, 11, 17, 15, ...] is convolved with [1, 2, 2, 2, 2, 2, 2, ...] we obtain A139250, the toothpick sequence. Example: A139250(5) = 15 = (1, 2, 2, 2, 2) * (3, 1, 3, 1, 1). - Gary W. Adamson, May 19 2009
Starting with 1 and convolved with [1, 2, 0, 0, 0, ...] = A151548. - Gary W. Adamson, Jun 04 2009
Refer to A162956 for the analogous triangle using N=3. - Gary W. Adamson, Jul 20 2009
It appears that the sums of two successive terms give the positive terms of A139251. - Omar E. Pol, Feb 18 2015

			a(2) = a(2^1+0) = 2*a(0) + a(1) = 1, a(3) = a(2^1+1) = 2*a(1) + a(2) = 3*a(2^i) = 2*a(0) + a(1) = 1.

N. J. A. Sloane, Table of n, a(n) for n = 0..16384
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

For the recurrence a(2^i+j) = C*a(j) + D*a(j+1), a(0) = A, a(1) = B for following values of (A B C D) see: (0 1 1 1) A118977, (1 0 1 1) A151702, (1 1 1 1) A151570, (1 2 1 1) A151571, (0 1 1 2) A151572, (1 0 1 2) A151703, (1 1 1 2) A151573, (1 2 1 2) A151574, (0 1 2 1) A160552, (1 0 2 1) A151704, (1 1 2 1) A151568, (1 2 2 1) A151569, (0 1 2 2) A151705, (1 0 2 2) A151706, (1 1 2 2) A151707, (1 2 2 2) A151708.
Cf. A152980, A139250, A139251, A151548, A160570, A151568.
Cf. A162956, A170903.

S:=proc(n) option remember; local i,j; if n <= 1 then RETURN(n); fi; i:=floor(log(n)/log(2)); j:=n-2^i; 2*S(j)+S(j+1); end; # N. J. A. Sloane, May 18 2009
H := x*(1+2*x)/(1+x) + (4*x^2/(1+2*x))*(mul(1+x^(2^k-1)+2*x^(2^k),k=1..20)-1); series(H,x,120); # N. J. A. Sloane, May 23 2009

Nest[Join[#, 2 # + Append[Rest@#, 1]] &, {0}, 7] (* Ivan Neretin, Feb 09 2017 *)

G.f.: x*(1+2*x)/(1+x) + (4*x^2/(1+2*x))*(-1 + Product_{k>=1} (1 + x^(2^k-1) + 2*x^(2^k))). - N. J. A. Sloane, May 23 2009, based on Gary W. Adamson's comment above and the known g.f. for A139250.
It appears that a(n) = A169708(n)/4, n >= 1. - Omar E. Pol, Feb 15 2015
It appears that a(n) = A139251(n) - a(n-1), n >= 1. - Omar E. Pol, Feb 18 2015

A169707 Total number of ON cells at stage n of two-dimensional cellular automaton defined by "Rule 750" using the von Neumann neighborhood.

Original entry on oeis.org

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, 1369, 1381, 1401, 1429, 1449, 1493, 1561, 1621, 1641, 1685, 1753, 1829, 1913, 2069, 2265, 2389, 2409, 2453, 2521

Offset: 1

N. J. A. Sloane, Apr 17 2010

Square grid, 4 neighbors per cell (N, E, S, W cells), turn ON iff exactly 1 or 3 neighbors are ON; once ON, cells stay ON.
The terms agree with those of A246335 for n <= 11, although the configurations are different starting at n = 7. - N. J. A. Sloane, Sep 21 2014
Offset 1 is best for giving a formula for a(n), although the Maple and Mathematica programs index the states starting at state 0.
It appears that this shares infinitely many terms with both A162795 and A147562, see Formula section and Example section. - Omar E. Pol, Feb 19 2015

			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)

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 928.

N. J. A. Sloane and Vincenzo Librandi, Table of n, a(n) for n = 1..513, computed from the definition, not using the conjectured formula. [The first 200 terms were found by _Vincenzo Librandi_]
David Applegate, The movie version
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.], arXiv:1004.3036 [math.CO].
N. J. A. Sloane, Table of n, a(n) for n=1..8192 assuming the recurrence is correct. Note that these terms are merely conjectures.
N. J. A. Sloane, Illustration of first 24 generations
N. J. A. Sloane, Illustration of first 24 generations of the NW-NE-SE-SW version of this CA
N. J. A. Sloane, Illustration of generation 7 of the NW-NE-SE-SW version of this CA (containing a(7) = 57 ON cells)
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata

Cf. A169708 (first differences), A147562, A147582, A169648, A169649, A169709, A169710, A246333, A246334, A246335, A246336, A253098 (partial sums).
Cf. also A162795, A150552, A160104, A170903, A048645, A187200.
See A253088 for the analogous CA using Rule 750 and a 9-celled neighborhood.

(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(2^k + i) = (4^(k+1)-1)/3 + 4*A246336(i), for k >= 0, 0 <= i < 2^k. For example, if n = 15 = 2^3 + 7, so k=3, i=7, we have a(15) = (4^4-1)/3 + 4*A246336(7) = 85 + 4*49 = 281.
a(n) = 1 + 2*(A139250(n) - A160552(n)) = A160164(n) - A170903(n) = A187220(n) + 2*(A160552(n-1)). - Omar E. Pol, Feb 18 2015
It appears that a(n) = A162795(n) = A147562(n), if n is a member of A048645, otherwise a(n) > A162795(n) > A147562(n). - Omar E. Pol, Feb 19 2015
It appears that a(n) = 1 + 4*A255747(n-1). - Omar E. Pol, Mar 05 2015
It appears that a(n) = 1 + 4*(A139250(n-1) - (a(n-1) - 1)/4), n > 1. - Omar E. Pol, Jul 24 2015
It appears that a(2n) = 1 + 4*A162795(n). - Omar E. Pol, Jul 04 2017

Edited (added formula, illustration, etc.) by N. J. A. Sloane, Aug 30 2014

Offset changed to 1 by N. J. A. Sloane, Feb 09 2015

A160164 Number of toothpicks after n-th stage in the I-toothpick structure of A139250.

Original entry on oeis.org

0, 2, 6, 14, 22, 30, 46, 70, 86, 94, 110, 134, 158, 190, 246, 310, 342, 350, 366, 390, 414, 446, 502, 566, 606, 638, 694, 766, 846, 966, 1142, 1302, 1366, 1374, 1390, 1414, 1438, 1470, 1526, 1590, 1630, 1662, 1718, 1790

Offset: 0

Omar E. Pol, Jun 01 2009

nonn

From Omar E. Pol, Mar 12 2011, Mar 15 2011, Mar 22 2011, Mar 25 2011: (Start)
We define an "I-toothpick" to consist of two connected toothpicks, as a bar of length 2. An I-toothpick with length 2 is formed by two toothpicks with length 1.
Note that in the physical model of the toothpick structure of A139250 the midpoint of a wooden toothpick of the new generation is superimposed on the endpoint of a wooden toothpick of the old generation. However, in the physical model of the I-toothpick structure the wooden toothpicks are not overlapping because all wooden toothpicks are connected by their endpoints.
a(n) is also the number of components after n-th stage in the toothpick structure of A139250, assuming the toothpicks have length 2.
Also, gullwing sequence starting from two opposite "gulls" (as a reflected gull in flight) such that the distance between their midpoints is equal to 2 (See A187220). The sequence gives the number of gulls in the structure after n-th stage.
Note that there is a correspondence between the gullwing structure and the I-toothpick structure, for example: a pair of opposite gulls in horizontal position in the gullwing structure is equivalent to a vertical I-toothpick with length 4 in the I-toothpick structure, such that the midpoint of each horizontal gull coincides with the midpoint of each vertical toothpick of the I-toothpick.
It appears this is also the connection between A147562 (the Ulam-Warburton cellular automaton) and the toothpick sequence A139250. The behavior of the function is similar to A147562 but here the structure is more complex. See Plot 2 button: A147562 vs A160164. See also A147562 vs A187220.
Also, B-toothpick sequence starting from two opposite "bells" such that the distance between their midpoints is equal to 4 (See A187220). We define a "B-toothpick" to consist of four arcs of length Pi/2 forming a "bell" similar to the Gauss function. A bell-shaped toothpick or B-toothpick or simply "bell" is formed by four Q-toothpicks (see A187210). A B-toothpick has length 2*Pi.  The sequence gives the number of bells in the structure after n stages.
We can see a correspondence between this structure and the I-toothpick structure of A139250. In this case, for example, a pair of opposite bells in horizontal position is equivalent to a vertical I-toothpick with length 8 in the I-toothpick structure, such that the midpoint of each horizontal bell coincides with the midpoint of each vertical toothpick of the I-toothpick.
Also, there is a fourth structure formed by isosceles right triangles, starting from two opposite triangles, since gulls or bells can be replaced by this type of triangles.
Note that the size of the toothpicks, gulls, bells and isosceles right triangles can be adjusted such that two or more of these structures can be overlaid.
(End)
The graph of this sequence is very close to the graphs of both A147562 and A169707 (see Plot 2). - Omar E. Pol, Feb 16 2015
It appears that a(n) is also the total number of ON cells after n-th stage in the half structure of the cellular automaton described in A169707 plus the total number of ON cells after n+1 stages in the half structure of the mentioned cellular automaton, without its central cell. See the illustration of the NW-NE-SE-SW version in A169707. - Omar E. Pol, Jul 26 2015
On the infinite Cairo pentagonal tiling consider the symmetric figure formed by two non-adjacent pentagons connected by a line segment joining two trivalent nodes. At stage 1 we start with one of these figures turned ON. a(n) is the number of ON cells in the structure after n-th stage, so a(1) = 2. The rule for the next stages is that the concave part of the figures of the new generation must be adjacent to the complementary convex part of the figures of the old generation. - Omar E. Pol, Mar 29 2018

			From _Omar E. Pol_, Aug 12 2013: (Start)
Illustration of initial terms:
.                                           _ _     _ _
.                     _ _ _ _   |_ _ _ _|    |_ _ _ _|
.       _ _   |_ _|    |_ _|    | |_ _| |   _|_|_ _|_|_
.   |    |    | | |    | | |      | | |        | | |
.   |   _|_   |_|_|    |_|_|      |_|_|     _ _|_|_|_ _
.             |   |   _|_ _|_   |_|_ _|_|    |_|_ _|_|
.                               |       |   _|_     _|_
.
.   2    6      14       22         30           46
.
(End)

N. J. A. Sloane, Table of n, a(n) for n = 0..16384
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.],
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Wikipedia, Cairo pentagonal tiling
Index entries for sequences related to cellular automata

Cf. A139250, A147562, A169707, A170903, A187210, A187220.

CoefficientList[Series[(2 x / ((1 - x) (1 + 2 x))) (1 + 2 x Product[1 + x^(2^k - 1) + 2 x^(2^k), {k, 0, 20}]), {x, 0, 53}], x] (* Vincenzo Librandi, Feb 15 2015 *)

a(n) = 2*A139250(n).
a(n) = A187220(n+1) - 1. - Omar E. Pol, Mar 12 2011, Mar 22 2011
It appears that a(n) = A169707(n) + A170903(n), n >= 1. - Omar E. Pol, Feb 15 2015
It appears that a(n) = (A169707(n) - 1)/2 + (A169707(n+1) - 1)/2, n >= 1. - Omar E. Pol, Jul 24 2015

Zero inserted, more terms and edited by Omar E. Pol, Mar 12 2011

A169708 First differences of A169707.

Original entry on oeis.org

1, 4, 4, 12, 4, 12, 20, 28, 4, 12, 20, 28, 20, 44, 68, 60, 4, 12, 20, 28, 20, 44, 68, 60, 20, 44, 68, 76, 84, 156, 196, 124, 4, 12, 20, 28, 20, 44, 68, 60, 20, 44, 68, 76, 84, 156, 196, 124, 20, 44, 68, 76, 84, 156, 196, 140, 84, 156, 212, 236, 324, 508, 516, 252, 4, 12, 20, 28, 20

Offset: 0

N. J. A. Sloane, Apr 17 2010

			From _Omar E. Pol_, Feb 13 2015: (Start)
Written as an irregular triangle in which row lengths are 1,1,2,4,8,16,32,... the sequence begins:
1;
4;
4,12;
4,12,20,28;
4,12,20,28,20,44,68,60;
4,12,20,28,20,44,68,60,20,44,68,76,84,156,196,124;
4,12,20,28,20,44,68,60,20,44,68,76,84,156,196,124,20,44,68,76,84,156,196,140,84,156,212,236,324,508,516,252;
It appears that the row sums give A000302.
It appears that the right border gives A173033.
(End)

David Applegate, The movie version
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata

Cf. A000302, A011782, A139251, A151548, A152980, A153006, A160164, A160552, A169707 (partial sums), A170903, A173033, A253088.

It appears that a(n) = 4*A160552(n), n >= 1. - Omar E. Pol, Feb 13 2015

Initial 1 added by Omar E. Pol, Feb 13 2015

A246336 Partial sums of A151548.

Original entry on oeis.org

1, 4, 9, 16, 21, 32, 49, 64, 69, 80, 97, 116, 137, 176, 225, 256, 261, 272, 289, 308, 329, 368, 417, 452, 473, 512, 565, 624, 705, 832, 961, 1024, 1029, 1040, 1057, 1076, 1097, 1136, 1185, 1220, 1241, 1280, 1333, 1392, 1473, 1600, 1729, 1796, 1817, 1856, 1909, 1968, 2049, 2176

Offset: 0

N. J. A. Sloane, Aug 30 2014

Arises in the analysis of a certain 2-D cellular automaton (see A169707).
a(46) = 1729 is also the Hardy-Ramanujan number. - Omar E. Pol, Feb 17 2015
It appears that sums of two successive terms give the numbers greater than 1 in A194811. - Omar E. Pol, Mar 05 2015

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata

Cf. A169707, A151548, A160552, A170903.

G.f.: 1/(1-x^2) + (4*x/(1-x))*mul(1+x^(2^k-1)+2*x^(2^k),k=1..oo).
From Omar E. Pol, Feb 18 2015: (Start)
It appears that:
a(2^k-2) = (2^k-1)^2, if k >= 1.
a(2^k-1) = 4^k, if k >= 1.
a(2^k) = 4^k + 5, if k >= 1.
(End)

A255166 Difference after n generations between the total number of single toothpicks in the I-toothpick structure of A160164 and the total number of ON cells in the "Ulam-Warburton" two-dimensional cellular automaton of A147562.

Original entry on oeis.org

0, 1, 1, 5, 1, 5, 9, 21, 1, 5, 9, 21, 9, 29, 49, 77, 1, 5, 9, 21, 9, 29, 49, 77, 9, 29, 49, 85, 57, 141, 209, 261, 1, 5, 9, 21, 9, 29, 49, 77, 9, 29, 49, 85, 57, 141, 209, 261, 9, 29, 49, 85, 57, 141, 209, 269, 57, 141, 217, 333, 289, 597, 785, 845, 1, 5, 9, 21, 9, 29, 49, 77, 9, 29, 49, 85, 57, 141, 209, 261, 9, 29, 49, 85

Offset: 0

Omar E. Pol, Feb 15 2015

			Written as an irregular triangle in which the row lengths are the terms of A011782, the sequence begins:
0;
1;
1,5;
1,5,9,21;
1,5,9,21,9,29,49,77;
1,5,9,21,9,29,49,77,9,29,49,85,57,141,209,261;
1,5,9,21,9,29,49,77,9,29,49,85,57,141,209,261,9,29,49,85,57,141,209,269,57,141,217,333,289,597,785,845;
...
It appears that the right border gives [0, 1] together with A126645. The right border gives the largest difference between both C.A. in every period.
Also, written the positive terms as an irregular triangle in which the row lengths are the terms of A011782, the sequence begins:
1;
1;
5,1;
5,9,21,1;
5,9,21,9,29,49,77,1;
5,9,21,9,29,49,77,9,29,49,85,57,141,209,261,1;
5,9,21,9,29,49,77,9,29,49,85,57,141,209,261,9,29,49,85,57,141,209,269,57,141,217,333,289,597,785,845,1;
...
The right border gives A000012 according with the illustrations as shown below. In this triangle the right border gives the smallest difference between both C.A. in every period.
For example: after 8 generations the structures look like this:
.
.                                      O
.                                    O O O
.                                  O   O   O
.    _ _ _ _ _ _ _ _             O O O O O O O
.     |_ _|   |_ _|            O   O   O   O   O
.     | |_|_ _|_| |          O O O   O O O   O O O
      |_|_|_ _|_|_|        O   O   O   O   O   O   O
.     |   | | |   |      O O O O O O O O O O O O O O O
.     |_ _|_|_|_ _|        O   O   O   O   O   O   O
.     | |_|_ _|_| |          O O O   O O O   O O O
.     |_|_|   |_|_|            O   O   O   O   O
.    _|_ _|_ _|_ _|_             O O O O O O O
.                                  O   O   O
.     86 toothpicks                  O O O
.                                      O
.
.                                 85 ON cells
.
a(8) = 1 because the I-toothpick structure contains 86 single toothpicks and the "Ulam-Warburton" two-dimensional cellular automaton has 85 ON cells, so the difference of the number of elements between both structures is equal to 86 - 85 = 1.

Cf. A126645, A139250, A147562, A160164, A169707, A170903.

a(n) = A160164(n) - A147562(n).

cp's OEIS Frontend

A160552 a(0)=0, a(1)=1; a(2^i+j) = 2*a(j) + a(j+1) for 0 <= j < 2^i.

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A169707 Total number of ON cells at stage n of two-dimensional cellular automaton defined by "Rule 750" using the von Neumann neighborhood.

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A160164 Number of toothpicks after n-th stage in the I-toothpick structure of A139250.

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A169708 First differences of A169707.

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A246336 Partial sums of A151548.

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A255166 Difference after n generations between the total number of single toothpicks in the I-toothpick structure of A160164 and the total number of ON cells in the "Ulam-Warburton" two-dimensional cellular automaton of A147562.

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