A162795 -id:A162795 - cp's OEIS frontend

A147562 Number of "ON" cells at n-th stage in the "Ulam-Warburton" two-dimensional cellular automaton.

0, 1, 5, 9, 21, 25, 37, 49, 85, 89, 101, 113, 149, 161, 197, 233, 341, 345, 357, 369, 405, 417, 453, 489, 597, 609, 645, 681, 789, 825, 933, 1041, 1365, 1369, 1381, 1393, 1429, 1441, 1477, 1513, 1621, 1633, 1669, 1705, 1813, 1849, 1957, 2065, 2389, 2401, 2437, 2473

Offset: 0

N. J. A. Sloane, based on emails from Franklin T. Adams-Watters, R. J. Mathar and David W. Wilson, Apr 29 2009

Studied by Holladay and Ulam circa 1960. See Fig. 1 and Example 1 of the Ulam reference. - N. J. A. Sloane, Aug 02 2009.
Singmaster calls this the Ulam-Warburton cellular automaton. - N. J. A. Sloane, Aug 05 2009
On the infinite square grid, start with all cells OFF.
Turn a single cell to the ON state.
At each subsequent step, each cell with exactly one neighbor ON is turned ON, and everything that is already ON remains ON.
Here "neighbor" refers to the four adjacent cells in the X and Y directions.
Note that "neighbor" could equally well refer to the four adjacent cells in the diagonal directions, since the graph formed by Z^2 with "one-step rook" adjacencies is isomorphic to Z^2 with "one-step bishop" adjacencies.
Also toothpick sequence starting with a central X-toothpick followed by T-toothpicks (see A160170 and A160172). The sequence gives the number of polytoothpicks in the structure after n-th stage. - Omar E. Pol, Mar 28 2011
It appears that this sequence shares infinitely many terms with both A162795 and A169707, see Formula section and Example section. - Omar E. Pol, Feb 20 2015
It appears that the positive terms are also the odd terms (a bisection) of A151920. - Omar E. Pol, Mar 06 2015
Also, the number of active (ON, black) cells in the n-th stage of growth of two-dimensional cellular automaton defined by Wolfram's "Rule 558" or "Rule 686" based on the 5-celled von Neumann neighborhood. - Robert Price, May 10 2016
From Omar E. Pol, Mar 05 2019: (Start)
a(n) is also the total number of "hidden crosses" after 4*n stages in the toothpick structure of A139250, including the central cross, beginning to count the crosses when their nuclei are totally formed with 4 quadrilaterals.
a(n) is also the total number of "flowers with six petals" after 4*n stages in the toothpick structure of A323650.
Note that the location of the "nuclei of the hidden crosses" and the "flowers with six petals" in both toothpick structures is essentially the same as the location of the "ON" cells in the version "one-step bishop" of this sequence (see the illustration of initial terms, figure 2). (End)
This sequence has almost exactly the same graph as A187220, A162795, A169707 and A160164 which is twice A139250. - Omar E. Pol, Jun 18 2022

			If we label the generations of cells turned ON by consecutive numbers we get a rosetta cell pattern:
. . . . . . . . . . . . . . . . .
. . . . . . . . 4 . . . . . . . .
. . . . . . . 4 3 4 . . . . . . .
. . . . . . 4 . 2 . 4 . . . . . .
. . . . . 4 3 2 1 2 3 4 . . . . .
. . . . . . 4 . 2 . 4 . . . . . .
. . . . . . . 4 3 4 . . . . . . .
. . . . . . . . 4 . . . . . . . .
. . . . . . . . . . . . . . . . .
In the first generation, only the central "1" is ON, a(1)=1. In the next generation, we turn ON four "2", leading to a(2)=a(1)+4=5. In the third generation, four "3" are turned ON, a(3)=a(2)+4=9. In the fourth generation, each of the four wings allows three 4's to be turned ON, a(4)=a(3)+4*3=21.
From _Omar E. Pol_, Feb 18 2015: (Start)
Also, written as an irregular triangle T(j,k), j>=0, k>=1, in which the row lengths are the terms of A011782:
1;
5;
9,   21;
25,  37, 49, 85;
89, 101,113,149,161,197,233,341;
345,357,369,405,417,453,489,597,609,645,681,789,825,933,1041,1365;
...
The right border gives the positive terms of A002450.
(End)
It appears that T(j,k) = A162795(j,k) = A169707(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, ... - _Omar E. Pol_, Feb 20 2015

S. Ulam, On some mathematical problems connected with patterns of growth of figures, pp. 215-224 of R. E. Bellman, ed., Mathematical Problems in the Biological Sciences, Proc. Sympos. Applied Math., Vol. 14, Amer. Math. Soc., 1962.
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 928.

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
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.]
Steven R. Finch, Toothpicks and Live Cells, July 21, 2015. [Cached copy, with permission of the author]
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 31.
Bradley Klee, Log-periodic coloring of the first quadrant, over the chair tiling.
Omar E. Pol, Illustration of initial terms (Fig. 1: one-step rook - the current sequence), (Fig. 2: one-step bishop), (Fig. 3: overlapping squares), (Fig. 4: overlapping X-toothpicks), (2009), (Fig. 5: overlapping circles), (2010)
Omar E. Pol, Illustration of initial terms of A139250, A160120, A147562 (overlapping figures), (2009).
David Singmaster, On the cellular automaton of Ulam and Warburton, M500 Magazine of the Open University, #195 (December 2003), pp. 2-7. Also scanned annotated cached copy, included with permission.
N. J. A. Sloane, Illustration of terms 0 through 9
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
N. J. A. Sloane, Exciting Number Sequences (video of talk), Mar 05 2021
N. J. A. Sloane and Brady Haran, Terrific Toothpick Patterns, Numberphile video (2018).
Mike Warburton, Ulam-Warburton Automaton - Counting Cells with Quadratics, arXiv:1901.10565 [math.CO], 2019.
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata

Cf. A000120, A139250, A147582 (number turned ON at n-th step), A147610, A130665, A151920, A151917, A160120, A160164, A160410, A160414, A162795, A169707, A187220, A246331, A323650.

A048645 Integers with one or two 1-bits in their binary expansion.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 17, 18, 20, 24, 32, 33, 34, 36, 40, 48, 64, 65, 66, 68, 72, 80, 96, 128, 129, 130, 132, 136, 144, 160, 192, 256, 257, 258, 260, 264, 272, 288, 320, 384, 512, 513, 514, 516, 520, 528, 544, 576, 640, 768, 1024, 1025, 1026, 1028, 1032

Offset: 1

Antti Karttunen, Jul 14 1999

Apart from initial 1, sums of two not necessarily distinct powers of 2.
4 does not divide C(2s-1,s) (= A001700[ s ]) if and only if s=a(n).
Possible number of sides of a regular polygon such that there exists a triangulation where each triangle is isosceles. - Sen-peng Eu, May 07 2008
Also numbers n such that n!/2^(n-2) is an integer. - Michel Lagneau, Mar 28 2011
It appears these are also the indices of the terms that are shared by the cellular automata of A147562, A162795, A169707. - Omar E. Pol, Feb 21 2015
Numbers with binary weight 1 or 2. - Omar E. Pol, Feb 22 2015

			From _Omar E. Pol_, Feb 18 2015: (Start)
Also, written as a triangle T(j,k), k >= 1, in which row lengths are the terms of A028310:
   1;
   2;
   3,  4;
   5,  6,  8;
   9, 10, 12, 16;
  17, 18, 20, 24, 32;
  33, 34, 36, 40, 48, 64;
  65, 66, 68, 72, 80, 96, 128;
  ...
It appears that column 1 is A094373.
It appears that the right border gives A000079.
It appears that the first differences in every row that contains at least two terms give the first h-1 powers of 2, where h is the length of the row.
(End)

Reinhard Zumkeller, Rows n = 1..100 of triangle, flattened
Wawrzyniec Bieniawski, Piotr Masierak, Andrzej Tomski, and Szymon Łukaszyk, Assembly Theory - Formalizing Assembly Spaces and Discovering Patterns and Bounds, Preprints.org (2025).
Michael P. Connolly, Probabilistic rounding error analysis for numerical linear algebra, Ph. D. Thesis, Univ. Manchester (UK, 2022). See p. 55.
USA Mathematical Olympiad, Problem 4, 2008.
Eric Weisstein's World of Mathematics, Automatic Set.
Eric Weisstein's World of Mathematics, Binomial Coefficient.
Index entries for sequences related to cellular automata.
Index to sequences related to Olympiads and other Mathematical competitions.

Cf. A018900, A048623, A046097, A169707, A147562, A162795, A003056, A002262, A094373, A028310, A179951.

import Data.List (insert)
a048645 n k = a048645_tabl !! (n-1) !! (k-1)
a048645_row n = a048645_tabl !! (n-1)
a048645_tabl = iterate (\xs -> insert (2 * head xs + 1) $ map ((* 2)) xs) [1]
a048645_list = concat a048645_tabl
-- Reinhard Zumkeller, Dec 19 2012

lincom:=proc(a,b,n) local i,j,s,m; s:={}; for i from 0 to n do for j from 0 to n do m:=a^i+b^j; if m<=n then s:={op(s),m} fi od; od; lprint(sort([op(s)])); end: lincom(2,2,1000); # Zerinvary Lajos, Feb 24 2007

Select[Range[2000], 1 <= DigitCount[#, 2, 1] <= 2&] (* Jean-François Alcover, Mar 06 2016 *)

isok(n) = my(hw = hammingweight(n)); (hw == 1) || (hw == 2); \\ Michel Marcus, Mar 06 2016

a(n) = if(n <= 2, return(n), n-=2); my(c = (sqrtint(8*n + 1) - 1) \ 2); 1 << c + 1 << (n - binomial(c + 1, 2)) \\ David A. Corneth, Jan 02 2019

nxt(n) = msb = 1 << logint(n, 2); if(n == msb, n + 1, t = n - msb; n + t) \\ David A. Corneth, Jan 02 2019

def ok(n): return 1 <= bin(n)[2:].count('1') <= 2
print([k for k in range(1033) if ok(k)]) # Michael S. Branicky, Jan 22 2022

from itertools import count, islice
def agen(): # generator of terms
    for d in count(0):
        msb = 2**d
        yield msb
        for lsb in range(d):
            yield msb + 2**lsb
print(list(islice(agen(), 60))) # Michael S. Branicky, Jan 22 2022

from math import isqrt, comb
def A048645(n): return (1<<(m:=isqrt(n-1<<3)+1>>1)-1)+(1<<(n-2-comb(m,2))) if n>1 else 1 # Chai Wah Wu, Oct 30 2024

a(0) = 1, a(n) = (2^(trinv(n-1)-1) + 2^((n-1)-((trinv(n-1)*(trinv(n-1)-1))/2))), i.e., 2^A003056(n) + 2^A002262(n-1) (the latter sequence contains the definition of trinv).
Let Theta = Sum_{k >= 0} x^(2^k). Then Sum_{n>=1} x^a(n) = (Theta^2 + Theta + x)/2. - N. J. A. Sloane, Jun 23 2009
As a triangle, for n > 1, 1 < k <= n: T(n,1) = A173786(n-2,n-2) and T(n,k) = A173786(n-1,k-2). - Reinhard Zumkeller, Feb 28 2010
It appears that A147562(a(n)) = A162795(a(n)) = A169707(a(n)). - Omar E. Pol, Feb 19 2015
Sum_{n>=1} 1/a(n) = 2 + A179951. - Amiram Eldar, Jan 22 2022

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

A160172 T-toothpick sequence (see Comments lines for definition).

Original entry on oeis.org

0, 1, 4, 9, 18, 27, 36, 49, 74, 95, 104, 117, 142, 167, 192, 229, 302, 359, 368, 381, 406, 431, 456, 493, 566, 627, 652, 689, 762, 835, 908, 1017, 1234, 1399, 1408, 1421, 1446, 1471, 1496, 1533, 1606, 1667, 1692, 1729, 1802, 1875, 1948, 2057, 2274, 2443, 2468

Offset: 0

Omar E. Pol, Jun 01 2009

A T-toothpick is formed from three toothpicks of equal length, in the shape of a T. There are three endpoints. We call the middle of the top toothpick the pivot point.
We start at round 0 with no T-toothpicks.
At round 1 we place a T-toothpick anywhere in the plane.
At round 2 we place three other T-toothpicks.
And so on...
The rule for adding a new T-toothpick is the following. A new T-toothpick is added at any exposed endpoint, with the pivot point touching the endpoint and so that the crossbar of the new toothpick is perpendicular to the exposed end.
The sequence gives the number of T-toothpicks after n rounds. A160173 (the first differences) gives the number added at the n-th round.
See the entry A139250 for more information about the toothpick process and the toothpick propagation.
On the infinite square grid a T-toothpick can be represented as a square polyedge with three components from a central point: two consecutive components on the same straight-line and a centered orthogonal component.
If the T-toothpick has three components then at the n-th round the structure is a polyedge with 3*a(n) components.
From Omar E. Pol, Mar 26 2011: (Start)
For formula and more information see the Applegate-Pol-Sloane paper, chapter 11, "T-shaped toothpicks". See also A160173.
Also, this sequence can be illustrated using another structure in which every T-toothpick is replaced by an isosceles right triangle. (End)
The structure is very distinct but the graph is similar to the graphs from the following sequences: A147562, A160164, A162795, A169707, A187220, A255366, A256260, at least for the known terms from Data section. - Omar E. Pol, Nov 24 2015
Shares with A255366 some terms with the same index, for example the element a(43) = 1729, the Hardy-Ramanujan number. - Omar E. Pol, Nov 25 2015

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.],
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A139250, A139251, A147562, A160120, A160160, A160164, A160170, A160173, A160406, A160408, A160426, A160800, A162795, A169707, A187220, A255366, A256260.

wt[n_] := DigitCount[n, 2, 1];
A151920[n_] := Sum[3^wt[i], {i, 1, n + 1}]/3;
a[n_] := 2*A151920[n - 2] + 2*A151920[n - 3] + n;
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 21 2024, after Charlie Neder *)

a(n) = 2*A151920(n) + 2*A151920(n-1) + n + 1. - Charlie Neder, Feb 07 2019

Edited and extended by N. J. A. Sloane, Jan 01 2010

A162796 Number of toothpicks in the toothpick structure A139250 that are orthogonal to the initial toothpick after n even rounds.

Original entry on oeis.org

0, 2, 6, 14, 22, 30, 42, 70, 86, 94, 106, 134, 154, 182, 222, 310, 342, 350, 362, 390, 410, 438, 478, 566, 602, 630, 670, 758, 814, 906, 1046, 1302, 1366, 1374, 1386, 1414, 1434, 1462, 1502, 1590, 1626, 1654, 1694, 1782, 1838, 1930, 2070, 2326, 2394, 2422, 2462

Offset: 0

Omar E. Pol, Jul 14 2009

nonn

Also, partial sums of A162794.

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

Cf. A139250, A139251, A159791, A159792, A162793, A162794, A162795, A162797.

a139251 := BFILETOLIST("b139251.txt") ; A162794 := proc(n) global a139251; op(2*n,a139251) ; end: A162796 := proc(n) add( A162794(k),k=1..n) ; end: seq(A162796(n),n=1..120) ; # R. J. Mathar, Sep 27 2009

terms = 100;
Cases[Import["https://oeis.org/A139251/b139251.txt", "Table"], {, }][[;; 2terms;; 2, 2]] // Accumulate (* Jean-François Alcover, Mar 24 2020 *)

Extended by R. J. Mathar, Sep 27 2009

A255747 Partial sums of A160552.

Original entry on oeis.org

0, 1, 2, 5, 6, 9, 14, 21, 22, 25, 30, 37, 42, 53, 70, 85, 86, 89, 94, 101, 106, 117, 134, 149, 154, 165, 182, 201, 222, 261, 310, 341, 342, 345, 350, 357, 362, 373, 390, 405, 410, 421, 438, 457, 478, 517, 566, 597, 602, 613, 630, 649, 670, 709, 758, 793, 814, 853, 906, 965, 1046, 1173, 1302, 1365, 1366, 1369, 1374

Offset: 0

Omar E. Pol, Mar 05 2015

nonn

It appears that the sums of two successive terms give the positive terms of the toothpick sequence A139250.
It appears that the odd terms (a bisection) give A162795.
It appears that a(n) is also the total number of ON cells at stage n+1 in one of the four wedges of two-dimensional cellular automaton "Rule 750" using the von Neumann neighborhood (see A169707). Therefore a(n) is also the total number of ON cells at stage n+1 in one of the four quadrants of the NW-NE-SE-SW version of that cellular automaton.
See also the formula section.
First differs from A169779 at a(11).

			Also, written as an irregular triangle in which the row lengths are the terms of A011782 (the number of compositions of n, n >= 0), the sequence begins:
0;
1;
2,   5;
6,   9, 14, 21;
22, 25, 30, 37, 42, 53, 70, 85;
86, 89, 94,101,106,117,134,149,154,165,182,201,222,261,310,341;
...
It appears that the first column gives 0 together with the terms of A047849, hence the right border gives A002450.
It appears that this triangle only shares with A151920 the positive elements of the columns 1, 2, 4, 8, 16, ... (the powers of 2).

Ivan Neretin, Table of n, a(n) for n = 0..8191
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
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. A002450, A011782, A047849, A139250, A151548, A151920, A160552, A162795, A169707, A169779, A246336.

Accumulate[Nest[Join[#, 2 # + Append[Rest@#, 1]] &, {0}, 6]] (* Ivan Neretin, Feb 09 2017 *)

It appears that a(n) + a(n-1) = A139250(n), n >= 1.
It appears that a(2n-1) = A162795(n), n >= 1.
It appears that a(n) = (A169707(n+1) - 1)/4.

A162797 a(n) = difference between the number of toothpicks of A139250 that are orthogonal to the initial toothpick and the number of toothpicks that are parallel to the initial toothpick, after n even rounds.

Original entry on oeis.org

1, 1, 5, 1, 5, 5, 17, 1, 5, 5, 17, 5, 17, 21, 49, 1, 5, 5, 17, 5, 17, 21, 49, 5, 17, 21, 49, 21, 53, 81, 129, 1, 5, 5, 17, 5, 17, 21, 49, 5, 17, 21, 49, 21, 53, 81, 129, 5, 17, 21, 49, 21, 53, 81, 129

Offset: 1

Omar E. Pol, Jul 14 2009

nonn

It appears that a(2^k) = 1, for k >= 0. [From Omar E. Pol, Feb 22 2010]

			Contribution from _Omar E. Pol_, Feb 22 2010: (Start)
If written as a triangle:
1;
1,5;
1,5,5,17;
1,5,5,17,5,17,21,49;
1,5,5,17,5,17,21,49,5,17,21,49,21,53,81,129;
1,5,5,17,5,17,21,49,5,17,21,49,21,53,81,129,5,17,21...
Rows converge to A173464.
(End)
Contribution from Omar E. Pol, Apr 01 2011 (Start):
It appears that the final terms of rows give A000337.
It appears that row sums give A006516.
(End)

Nathaniel Johnston, Table of n, a(n) for n = 1..94
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

Cf. A139250, A139251, A159791, A159792, A162793, A162794, A162795, A162796.

Cf. A000337, A058922, A173464. [From Omar E. Pol, Feb 22 2010]

a(n) = A162796(n) - A162795(n).

Edited by Omar E. Pol, Jul 18 2009
More terms from Omar E. Pol, Feb 22 2010
More terms (a(51)-a(55)) from Nathaniel Johnston, Mar 30 2011

cp's OEIS Frontend

A261895 Decimal expansion of the lower limit of A162795(i)/i^2.

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A261896 Consider the 2^n values of A162795(i)/i^2 for 2^n <= i < 2^(n+1); a(n) = value of i where this quantity is minimized.

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A147562 Number of "ON" cells at n-th stage in the "Ulam-Warburton" two-dimensional cellular automaton.

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A048645 Integers with one or two 1-bits in their binary expansion.

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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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A160172 T-toothpick sequence (see Comments lines for definition).

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