Displaying 1-10 of 18 results found.
1, 8, 12, 28, 12, 36, 36, 92, 12, 36, 36, 108, 36, 108, 108, 292, 12, 36, 36, 108, 36, 108, 108, 324, 36, 108, 108, 324, 108, 324, 324, 908, 12, 36, 36, 108, 36, 108, 108, 324, 36, 108, 108, 324, 108, 324, 324, 972, 36, 108, 108, 324, 108, 324, 324, 972, 108, 324, 324
MAPLE
isA000079 := proc(n) if type(n, 'even') then nops(numtheory[factorset](n)) = 1 ; else false ; fi ; end proc:
A048883 := proc(n) 3^wt(n) ; end proc:
A161415 := proc(n) if n = 1 then 1; elif isA000079(n) then 4* A048883(n-1)-2*n ; else 4* A048883(n-1) ; end if; end proc: seq( A161415(n), n=1..90) ; (End)
MATHEMATICA
a[1] = 1; a[n_] := 4*3^DigitCount[n-1, 2, 1] - If[IntegerQ[Log[2, n]], 2n, 0];
Number of "ON" cells at n-th stage of three-dimensional version of the cellular automaton A160414 using cubes.
+20
5
0, 1, 27, 83, 343, 399, 791, 1183, 3375, 3431, 3823, 4215, 6959, 7351, 10095, 12839, 29791, 29847, 30239, 30631, 33375, 33767, 36511, 39255, 58463, 58855, 61599, 64343, 83551, 86295, 105503, 124711, 250047, 250103, 250495, 250887, 253631, 254023, 256767, 259511
COMMENTS
For the first differences see A161341, where an explicit formula is given.
FORMULA
a(n) = (2n-1)^3 if n is a power of 2.
0, 2, 5, 12, 15, 24, 33, 56, 59, 68, 77, 104, 113, 140, 167, 240, 243, 252, 261, 288, 297, 324, 351, 432, 441, 468, 495, 576, 603, 684, 765, 992, 995, 1004, 1013, 1040, 1049, 1076, 1103, 1184, 1193, 1220, 1247, 1328, 1355, 1436
COMMENTS
As is obvious from symmetry, for all n>=1 one has A160414(n) = 1 + a multiple of 4. Partial sums of 3^ A000120(n-1)-(n/2 if n=2^k), n>1.
MATHEMATICA
A219954list[nmax_]:=Accumulate[Table[If[n==1, 0, 3^DigitCount[n-1, 2, 1]-If[IntegerQ[Log2[n]], n/2, 0]], {n, nmax}]]; A219954list[100] (* Paolo Xausa, Sep 01 2023 *)
PROG
(PARI) my(s=-1, t(n)=3^norml2(binary(n-1))-if(n==(1<<valuation(n, 2)), n\2)); vector(99, i, s+=t(i))
0, 1, 10, 31, 80, 141, 238, 371, 596, 833, 1106, 1415, 1832, 2285, 2846, 3515, 4476, 5449, 6458, 7503, 8656, 9845, 11142, 12547, 14276, 16041, 17914, 19895, 22200, 24613, 27350, 30411, 34380, 38361, 42378, 46431, 50592, 54789, 59094, 63507, 68244, 73017, 77898, 82887, 88200, 93621, 99366
Number of "ON" cells at n-th stage in the "Ulam-Warburton" two-dimensional cellular automaton.
+10
91
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
COMMENTS
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 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)
REFERENCES
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.
FORMULA
a(n) = 1 + 4*Sum_{k=1..n-1} 3^(wt(k)-1) for n>1, where wt() = A000120(). [Corrected by Paolo Xausa, Aug 12 2022] a(n) = 2* A151917(n) - 1, for n >= 1. a(n) = 1 + 4* A151920(n-2), for n >= 2. (End)
EXAMPLE
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.
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
MAPLE
Since this is the partial sum sequence of A147582, it is most easily obtained using the Maple code given in A147582. # [x, y] coordinates of cells on
Lse := [[0, 0]] ;
# enclosing rectangle of the cells on (that is, minima and maxima in Lse)
xmin := 0 ;
xmax := 0 ;
ymin := 0 ;
ymax := 0 ;
# count neighbors of x, y which are on; return 0 if [x, y] is in L
cntnei := proc(x, y, L)
local a, p, xpt, ypt;
a := 0 ;
if not [x, y] in L then
for p in Lse do
xpt := op(1, p) ;
ypt := op(2, p) ;
if ( abs(xpt-x) = 1 and ypt=y ) or ( x=xpt and abs(ypt-y) = 1) then
a := a+1 ;
fi;
od:
fi:
RETURN(a) ;
end:
# loop over generations/steps
for stp from 1 to 10 do
Lnew := [] ;
for x from xmin-1 to xmax+1 do
for y from ymin-1 to ymax+1 do
if cntnei(x, y, Lse) = 1 then
Lnew := [op(Lnew), [x, y]] ;
fi;
od:
od:
for p in Lnew do
xpt := op(1, p) ;
ypt := op(2, p) ;
xmin := min(xmin, xpt) ;
xmax := max(xmax, xpt) ;
ymin := min(ymin, ypt) ;
ymax := max(ymax, ypt) ;
od:
Lse := [op(Lse), op(Lnew)] ;
print(nops(Lse)) ;
MATHEMATICA
Join[{0}, Map[Function[Apply[Plus, Flatten[ #1]]], CellularAutomaton[{686, {2, {{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}}, {1, 1}}, {{{1}}, 0}, 200]]] (* Nadia Heninger and N. J. A. Sloane, Aug 11 2009; modified by Paolo Xausa, Aug 12 2022 to include the a(0) term *) ArrayPlot /@ CellularAutomaton[{686, {2, {{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}}, {1, 1}}, {{{1}}, 0}, 16] (* N. J. A. Sloane, Nov 08 2014 *) A147562list[nmax_]:=Accumulate[Join[{0, 1}, 4*3^(DigitCount[Range[nmax-1], 2, 1]-1)]]; A147562list[100] (* Paolo Xausa, May 21 2023 *)
PROG
(PARI) a(n) = if (n, 1 + 4*sum(k=1, n-1, 3^(hammingweight(k)-1)), 0); \\ Michel Marcus, Jul 05 2022
CROSSREFS
Cf. A000120, A139250, A147582 (number turned ON at n-th step), A147610, A130665, A151920, A151917, A160120, A160164, A160410, A160414, A162795, A169707, A187220, A246331, A323650.
EXTENSIONS
Numbers in the comment adapted to the offset by R. J. Mathar, Mar 03 2010
Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (see Comments for precise definition).
+10
22
0, 4, 16, 28, 64, 76, 112, 148, 256, 268, 304, 340, 448, 484, 592, 700, 1024, 1036, 1072, 1108, 1216, 1252, 1360, 1468, 1792, 1828, 1936, 2044, 2368, 2476, 2800, 3124, 4096, 4108, 4144, 4180, 4288, 4324, 4432, 4540, 4864, 4900, 5008, 5116, 5440, 5548, 5872, 6196
COMMENTS
On the infinite square grid, we consider cells to be the squares, and we start at round 0 with all cells in the OFF state, so a(0) = 0.
At round 1, we turn ON four cells, forming a square.
The rule for n > 1: A cell in turned ON iff exactly one of its four vertices is a corner vertex of the set of ON cells. So in each generation every exposed vertex turns on three new cells.
Therefore:
At Round 2, we turn ON twelve cells around the square.
At round 3, we turn ON twelve other cells. Three cells around of every corner of the square.
And so on.
For the first differences see the entry A161411. Shows a fractal behavior similar to the toothpick sequence A139250. A very similar sequence is A160414, which uses the same rule but with a(1) = 1, not 4. When n=2^k then the polygon formed by ON cells is a square with side length 2^(k+1).
a(n) is also the area of the figure of A147562 after n generations if A147562 is drawn as overlapping squares. - Omar E. Pol, Nov 08 2009 Also, toothpick sequence starting with four toothpicks centered at (0,0) as a cross.
Rule: Each exposed endpoint of the toothpicks of the old generation must be touched by the endpoints of three toothpicks of new generation. (Note that these three toothpicks looks like a T-toothpick, see A160172.) The sequence gives the number of toothpicks after n stages. A161411 gives the number of toothpicks added at the n-th stage. (End)
FORMULA
a(2^k) = (2*(2^k))^2 for k>=0.
EXAMPLE
With the positive terms written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins: 4;
16;
28, 64;
76, 112, 148, 256;
268, 304, 340, 448, 484, 592, 700, 1024;
...
Right border gives the elements of A000302 greater than 1. This triangle T(n,k) shares with the triangle A256534 the terms of the column k, if k is a power of 2, for example, both triangles share the following terms: 4, 16, 28, 64, 76, 112, 256, 268, 304, 448, 1024, etc. .
Illustration of initial terms, for n = 1..10:
. _ _ _ _ _ _ _ _
. | _ _ | | _ _ |
. | | _|_|_ _ _ _ _ _ _ _ _ _ _ _|_|_ | |
. | |_| _ _ _ _ _ _ _ _ |_| |
. |_ _| | _|_ _|_ | | _|_ _|_ | |_ _|
. | |_| _ _ |_| |_| _ _ |_| |
. | | | _|_|_ _ _ _|_|_ | | |
. | _| |_| _ _ _ _ |_| |_ |
. | | |_ _| | _|_ _|_ | |_ _| | |
. | |_ _| | |_| _ _ |_| | |_ _| |
. | | | | | | | |
. | _ _ | _| |_ _| |_ | _ _ |
. | | _|_| | |_ _ _ _| | |_|_ | |
. | |_| _| |_ _| |_ _| |_ |_| |
. | | | |_ _ _ _ _ _ _ _| | | |
. | _| |_ _| |_ _| |_ _| |_ |
. _ _| | |_ _ _ _| | | |_ _ _ _| | |_ _
. | _| |_ _| |_ _| |_ _| |_ _| |_ |
. | | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
. | |_ _| | | |_ _| |
. |_ _ _ _| |_ _ _ _|
.
After 10 generations there are 304 ON cells, so a(10) = 304.
(End)
MATHEMATICA
RasterGraphics[state_?MatrixQ, colors_Integer:2, opts___]:=
Graphics[Raster[Reverse[1-state/(colors -1)]],
AspectRatio ->(AspectRatio/.{opts}/.AspectRatio ->Automatic),
Frame ->True, FrameTicks ->None, GridLines ->None];
rule=1340761804646523638425234105559798690663900360577570370705802859623\
705267234688669629039040624964794287326910250673678735142700520276191850\
5902735959769690
Show[GraphicsArray[Map[RasterGraphics, CellularAutomaton[{rule, {2,
{{4, 2, 1}, {32, 16, 8}, {256, 128, 64}}}, {1, 1}}, {{{1, 1}, {1, 1}}, 0}, 9, -10]]]];
ca=CellularAutomaton[{rule, {2, {{4, 2, 1}, {32, 16, 8}, {256, 128, 64}}}, {1,
1}}, {{{1, 1}, {1, 1}}, 0}, 99, -100];
Table[Total[ca[[i]], 2], {i, 1, Length[ca]}]
a[n_] := 4*Sum[3^DigitCount[k, 2, 1], {k, 0, n-1}];
CROSSREFS
Cf. A000079, A000302, A048883, A139250, A147562, A160118, A160412, A160414, A161411, A160717, A160720, A160727, A256530, A256534.
Number of "ON" cells at n-th stage in 2-dimensional cellular automaton (see Comments for precise definition).
+10
17
0, 1, 5, 9, 21, 25, 37, 49, 77, 81, 93, 105, 133, 145, 173, 201, 261, 265, 277, 289, 317, 329, 357, 385, 445, 457, 485, 513, 573, 601, 661, 721, 845, 849, 861, 873, 901, 913, 941, 969, 1029, 1041, 1069, 1097, 1157, 1185, 1245, 1305, 1429, 1441, 1469, 1497
COMMENTS
We work on the vertices of the square grid Z^2, and define the neighbors of a cell to be the four closest cells along the diagonals.
We start at stage 0 with all cells in OFF state.
At stage 1, we turn ON a single cell at the origin.
Once a cell is ON it stays ON.
At each subsequent stage, a cell in turned ON if exactly one of its neighboring cells that are no further from the origin is ON.
The "no further from the origin" condition matters for the first time at stage 8, when only A160721(8) = 28 cells are turned ON, and a(8) = 77. In contrast, A147562(8) = 85, A147582(8) = 36. This CA also arises as the cross-section in the (X,Y)-plane of the CA in A151776. In other words, a cell is turned ON if exactly one of its vertices touches an exposed vertex of a ON cell of the previous generation. A special rule for this sequence is that every ON cell has only one vertex that should be considered not exposed: its nearest vertex to the center of the structure.
Analog to the "outward" version ( A266532) of the Y-toothpick cellular automaton of A160120 on the triangular grid, but here we have ON cells on the square grid. See also the formula section. - Omar E. Pol, Jan 19 2016 This cellular automaton can be interpreted as the outward version of the Ulam-Warburton two-dimensional cellular automaton (see A147562). - Omar E. Pol, Jun 22 2017
EXAMPLE
If we label the generations of cells turned ON by consecutive numbers we get the cell pattern shown below:
9...............9
.8.8.8.8.8.8.8.8.
..7...7...7...7..
.8.6.6.....6.6.8.
....5.......5....
.8.6.4.4.4.4.6.8.
..7...3...3...7..
.8...4.2.2.4...8.
........1........
.8...4.2.2.4...8.
..7...3...3...7..
.8.6.4.4.4.4.6.8.
....5.......5....
.8.6.6.....6.6.8.
..7...7...7...7..
.8.8.8.8.8.8.8.8.
9...............9
MAPLE
cellOn := [[0, 0]] : bbox := [0, 0, 0, 0]: # llx, lly, urx, ury isOn := proc(x, y, L) local i ; for i in L do if op(1, i) = x and op(2, i) = y then RETURN(true) ; fi; od: RETURN(false) ; end: bb := proc(L) local mamin, i; mamin := [0, 0, 0, 0] ; for i in L do mamin := subsop(1=min(op(1, mamin), op(1, i)), mamin) ; mamin := subsop(2=min(op(2, mamin), op(2, i)), mamin) ; mamin := subsop(3=max(op(1, mamin), op(1, i)), mamin) ; mamin := subsop(4=max(op(2, mamin), op(2, i)), mamin) ; od: mamin ; end: for gen from 2 to 80 do nGen := [] ; print(nops(cellOn)) ; for x from op(1, bbox)-1 to op(3, bbox)+1 do for y from op(2, bbox)-1 to op(4, bbox)+1 do # not yet in list? if not isOn(x, y, cellOn) then
# loop over 4 neighbors of (x, y) non := 0 ; for dx from -1 to 1 by 2 do for dy from -1 to 1 by 2 do # test of a neighbor nearer to origin if x^2+y^2 >= (x+dx)^2+(y+dy)^2 then if isOn(x+dx, y+dy, cellOn) then non := non+1 ; fi; fi; od: od: # exactly one neighbor on: add to nGen if non = 1 then nGen := [op(nGen), [x, y]] ; fi; fi; od: od: # merge old and new generation cellOn := [op(cellOn), op(nGen)] ; bbox := bb(cellOn) ; od: # R. J. Mathar, Jul 14 2009
MATHEMATICA
A160720[0]=0; A160720[n_]:=Total[With[{m = n - 1}, BitOr @@ (Function[pos, CellularAutomaton[{FromDigits[Boole[#[[2, 2]] == 1 || Count[Flatten[#], 1] == 1 && Count[Extract[#, pos], 1] == 1] & /@ Tuples[{1, 0}, {3, 3}], 2], 2, {1, 1}}, {{{1}}, 0}, {{{m}}, {-m, m}, {-m, m}}]] /@ Partition[{{-1, -1}, {-1, 1}, {1, 1}, {1, -1}}, 2, 1, 1])], 2] (* JungHwan Min, Jan 23 2016 *)
A160720[0]=0; A160720[n_]:=Total[With[{m = n - 1}, BitOr @@ (CellularAutomaton[{#, 2, {1, 1}}, {{{1}}, 0}, {{{m}}, {-m, m}, {-m, m}}] & /@ {1340760334615130450764733360212427074493015729158098619714804343768786376359766200271125675579697244361343863555105588 9478487182262900810351549134401372178, 13407603346151304507647333602124270744930157291580986197148043437687863763597777794800494071992396014598447323458909159463152822826940267935557047531012112, 13407603346151304507647333602124270744930157291580986197148043437687863763597777794800494071992396014598447323458909159463152822826940286382301121240563712, 13407603346151304507647333602124270744930157291580986197148043437687863763597662002711256755796972443613438635551055889478487182262900828798293208110923778})], 2] (* JungHwan Min, Jan 23 2016 *)
A160720[0]=0; A160720[n_]:=Total[With[{m = n - 1}, BitOr @@ (CellularAutomaton[{46, {2, ReplacePart[ArrayPad[{{1}}, 1], # -> 2]}, {1, 1}}, {{{1}}, 0}, {{{m}}, All, All}] & /@ Partition[{{-1, -1}, {-1, 1}, {1, 1}, {1, -1}}, 2, 1, 1])], 2] (* JungHwan Min, Jan 24 2016 *)
CROSSREFS
Cf. A139250, A147562, A159912, A160117, A160118, A160120, A160721, A160740, A147562, A160410, A160414, A160412, A160416, A160796, A266532, A267700.
a(n) = (Sum_{i=1..n+1} 3^wt(i))/3, where wt() = A000120().
+10
15
1, 2, 5, 6, 9, 12, 21, 22, 25, 28, 37, 40, 49, 58, 85, 86, 89, 92, 101, 104, 113, 122, 149, 152, 161, 170, 197, 206, 233, 260, 341, 342, 345, 348, 357, 360, 369, 378, 405, 408, 417, 426, 453, 462, 489, 516, 597, 600, 609, 618, 645, 654, 681, 708, 789, 798, 825, 852, 933, 960
COMMENTS
Partial sums of A147610 (but with offset changed to 0). It appears that the first bisection gives the positive terms of A147562. - Omar E. Pol, Mar 07 2015
EXAMPLE
n=3: (3^1+3^1+3^2+3^1)/3 = 18/3 = 6.
n=18: the binary expansion of 18+1 is 10011, i.e., 19 = 2^4 + 2^1 + 2^0.
The exponents of these powers of 2 (4, 1 and 0) reoccur as exponents in the powers of 4: a(19) = 3^0 * [(4^4 - 1) / 3 + 1] + 3^1 * [(4^1 - 1) / 3 + 1] + 3^2 * [(4^0 - 1)/3 + 1] = 1 * 86 + 3 * 2 + 9 * 1 = 101. - David A. Corneth, Mar 21 2015
MATHEMATICA
t = Nest[Join[#, # + 1] &, {0}, 14]; Table[Sum[3^t[[i + 1]], {i, 1, n}]/3, {n, 60}] (* Michael De Vlieger, Mar 21 2015 *)
PROG
(PARI) a(n) = sum(i=1, n+1, 3^hammingweight(i))/3; \\ Michel Marcus, Mar 07 2015 (PARI) a(n)={b=binary(n+1); t=#b; e=-1; sum(i=1, #b, e+=(b[i]==1); (b[i]==1)*3^e*((4^(#b-i)-1)/3+1))} \\ David A. Corneth, Mar 21 2015
4, 12, 12, 36, 12, 36, 36, 108, 12, 36, 36, 108, 36, 108, 108, 324, 12, 36, 36, 108, 36, 108, 108, 324, 36, 108, 108, 324, 108, 324, 324, 972, 12, 36, 36, 108, 36, 108, 108, 324, 36, 108, 108, 324, 108, 324, 324, 972, 36, 108, 108, 324, 108, 324, 324, 972, 108, 324, 324
COMMENTS
The rows of the triangle in A147582 converge to this sequence. Contribution from Omar E. Pol, Mar 28 2011 (Start):
a(n) is the number of cells turned "ON" at n-th stage of the cellular automaton of A160410. a(n) is also the number of toothpicks added at n-th stage to the toothpick structure of A160410. (End)
EXAMPLE
If written as a triangle:
.4;
.12;
.12,36;
.12,36,36,108;
.12,36,36,108,36,108,108,324;
MATHEMATICA
4*3^DigitCount[Range[0, 100], 2, 1] (* Paolo Xausa, Sep 01 2023 *)
Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (see Comments for precise definition).
+10
8
0, 3, 12, 21, 48, 57, 84, 111, 192, 201, 228, 255, 336, 363, 444, 525, 768, 777, 804, 831, 912, 939, 1020, 1101, 1344, 1371, 1452, 1533, 1776, 1857, 2100, 2343, 3072, 3081, 3108, 3135, 3216, 3243, 3324, 3405, 3648, 3675, 3756, 3837, 4080, 4161, 4404, 4647
COMMENTS
On the infinite square grid, consider the outside corner of an infinite square.
We start at round 0 with all cells in the OFF state.
The rule: A cell in turned ON iff exactly one of its four vertices is a corner vertex of the set of ON cells. So in each generation every exposed vertex turns on three new cells.
At round 1, we turn ON three cells around the corner of the infinite square, forming a concave-convex hexagon with three exposed vertices.
At round 2, we turn ON nine cells around the hexagon.
At round 3, we turn ON nine other cells. Three cells around of every corner of the hexagon.
And so on.
Shows a fractal-like behavior similar to the toothpick sequence A153006. For the first differences see the entry A162349. For more information see A160410, which is the main entry for this sequence. (End)
FORMULA
a(0) = 0, a(n) = A130665(n-1)*3, for n>0. (End)
EXAMPLE
If we label the generations of cells turned ON by consecutive numbers we get the cell pattern shown below:
...77..77..77..77
...766667..766667
....6556....6556.
....654444444456.
...76643344334667
...77.43222234.77
......44211244...
00000000001244...
00000000002234.77
00000000004334667
0000000000444456.
0000000000..6556.
0000000000.766667
0000000000.77..77
0000000000.......
0000000000.......
0000000000.......
MATHEMATICA
a[n_] := 3*Sum[3^DigitCount[k, 2, 1], {k, 0, n - 1}]; Array[a, 48, 0] (* Michael De Vlieger, Nov 01 2022 *)
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