cp's OEIS Frontend

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.

Showing 1-10 of 14 results. Next

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

Original entry on oeis.org

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

Views

Author

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

Keywords

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
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

Examples

			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

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.

Links

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.
See also A006046, A335794, A335795.

Programs

  • 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 *)
  • PARI
    a(n) = if (n, 1 + 4*sum(k=1, n-1, 3^(hammingweight(k)-1)), 0); \\ Michel Marcus, Jul 05 2022

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]
For asymptotics see the discussion in the comments in A006046. - N. J. A. Sloane, Mar 11 2021
From Omar E. Pol, Mar 13 2011: (Start)
a(n) = 2*A151917(n) - 1, for n >= 1.
a(n) = 1 + 4*A151920(n-2), for n >= 2.
(End)
It appears that a(n) = A162795(n) = A169707(n), if n is a member of A048645, otherwise a(n) < A162795(n) < A169707(n). - Omar E. Pol, Feb 20 2015
It appears that a(n) = A151920(2n-2), n >= 1. - Omar E. Pol, Mar 06 2015
It appears that a(n) = (A130665(2n-1) - 1)/3, n >= 1. - Omar E. Pol, Mar 07 2015
a(n) = 1 + 4*(A130665(n-1) - 1)/3, n >= 1. Omar E. Pol, Mar 07 2015
a(n) = A323650(2n)/3. - Omar E. Pol, Mar 04 2019

Extensions

Offset and initial terms changed by N. J. A. Sloane, Jun 07 2009
Numbers in the comment adapted to the offset by R. J. Mathar, Mar 03 2010

A048883 a(n) = 3^wt(n), where wt(n) = A000120(n).

Original entry on oeis.org

1, 3, 3, 9, 3, 9, 9, 27, 3, 9, 9, 27, 9, 27, 27, 81, 3, 9, 9, 27, 9, 27, 27, 81, 9, 27, 27, 81, 27, 81, 81, 243, 3, 9, 9, 27, 9, 27, 27, 81, 9, 27, 27, 81, 27, 81, 81, 243, 9, 27, 27, 81, 27, 81, 81, 243, 27, 81, 81, 243, 81, 243, 243, 729, 3, 9, 9, 27, 9, 27, 27, 81, 9, 27, 27, 81, 27, 81
Offset: 0

Views

Author

John W. Layman

Keywords

Comments

Or, a(n)=number of 1's ("live" cells) at stage n of a 2-dimensional cellular automata evolving by the rule: 1 if NE+NW+S=1, else 0.
This is the odd-rule cellular automaton defined by OddRule 013 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link). - N. J. A. Sloane, Feb 25 2015
Or, start with S=[1]; replace S by [S, 3*S]; repeat ad infinitum.
Fixed point of the morphism 1 -> 13, 3 -> 39, 9 -> 9(27), ... = 3^k -> 3^k 3^(k+1), ... starting from a(0) = 1; 1 -> 13 -> 1339 -> = 1339399(27) -> 1339399(27)399(27)9(27)(27)(81) -> ..., . - Robert G. Wilson v, Jan 24 2006
Equals row sums of triangle A166453 (the square of Sierpiński's gasket, A047999). - Gary W. Adamson, Oct 13 2009
First bisection of A169697=1,5,3,19,3,. a(2n+2)+a(2n+3)=12,12,36,=12*A147610 ? Distribution of terms (in A000244): A011782=1,A000079 for first array, A000079 for second. - Paul Curtz, Apr 20 2010
a(A000225(n)) = A000244(n) and a(m) != A000244(n) for m < A000225(n). - Reinhard Zumkeller, Nov 14 2011
This sequence pertains to phenotype Punnett square mathematics. Start with X=1. Each hybrid cross involves the equation X:3X. Therefore, the ratio in the first (mono) hybrid cross is X=1:3X=3(1) or 3; or 3:1. When you move up to the next hybridization level, replace the previous cross ratio with X. X now represents 2 numbers-1:3. Therefore, the ratio in the second (di) hybrid cross is X=(1:3):3X=[3(1):3(3)] or (3:9). Put it together and you get 1:3:3:9. Each time you move up a hybridization level, replace the previous ratio with X, and use the same equation-X:3X to get its ratio. - John Michael Feuk, Dec 10 2011
Number of odd values in the n-th layer of Pascal's tetrahedron (see A268240). - Caden Le, Mar 03 2025
a(x*y) <= a(x)^A000120(y). - Joe Amos, Mar 28 2025

Examples

			From _Omar E. Pol_, Jun 07 2009: (Start)
Triangle begins:
  1;
  3;
  3,9;
  3,9,9,27;
  3,9,9,27,9,27,27,81;
  3,9,9,27,9,27,27,81,9,27,27,81,27,81,81,243;
  3,9,9,27,9,27,27,81,9,27,27,81,27,81,81,243,9,27,27,81,27,81,81,243,27,...
Or
  1;
  3,3;
  9,3,9,9;
  27,3,9,9,27,9,27,27;
  81,3,9,9,27,9,27,27,81,9,27,27,81,27,81,81;
  243,3,9,9,27,9,27,27,81,9,27,27,81,27,81,81,243,9,27,27,81,27,81,81,243,27...
(End)

Links

Crossrefs

For generating functions Product_{k>=0} (1+a*x^(b^k)) for the following values of (a,b) see: (1,2) A000012 and A000027, (1,3) A039966 and A005836, (1,4) A151666 and A000695, (1,5) A151667 and A033042, (2,2) A001316, (2,3) A151668, (2,4) A151669, (2,5) A151670, (3,2) A048883, (3,3) A117940, (3,4) A151665, (3,5) A151671, (4,2) A102376, (4,3) A151672, (4,4) A151673, (4,5) A151674.
A generalization of A001316. Cf. A102376.
Partial sums give A130665. - David Applegate, Jun 11 2009
Cf. A000079, A122018, A166453, A268240.

Programs

  • Haskell
    a048883 = a000244 . a000120  -- Reinhard Zumkeller, Nov 14 2011
  • Mathematica
    Nest[ Join[#, 3#] &, {1}, 6] (* Robert G. Wilson v, Jan 24 2006 and modified Jul 27 2014*)
a[n_] := 3^DigitCount[n, 2, 1]; Array[a, 80, 0] (* Jean-François Alcover, Nov 15 2017 *)
  • PARI
    a(n)=n=binary(n);3^sum(i=1,#n,n[i])

Formula

a(n) = Product_{k=0..log_2(n)} 3^b(n,k), where b(n,k) = coefficient of 2^k in binary expansion of n (offset 0). - Paul D. Hanna
a(n) = 3*a(n/2) if n is even, otherwise a(n) = a((n+1)/2).
G.f.: Product_{k>=0} (1+3*x^(2^k)). The generalization k^A000120 has generating function (1 + kx)*(1 + kx^2)*(1 + kx^4)*...
a(n+1) = Sum_{i=0..n} (binomial(n, i) mod 2) * Sum_{j=0..i} (binomial(i, j) mod 2). - Benoit Cloitre, Nov 16 2003
a(0)=1, a(n) = 3*a(n-A053644(n)) for n > 0. - Joe Slater, Jan 31 2016
G.f. A(x) satisfies: A(x) = (1 + 3*x) * A(x^2). - Ilya Gutkovskiy, Jul 09 2019

Extensions

Corrected by Ralf Stephan, Jun 19 2003
Entry revised by N. J. A. Sloane, May 30 2009
Offset changed to 0, Jun 11 2009

A147582 First differences of A147562.

Original entry on oeis.org

1, 4, 4, 12, 4, 12, 12, 36, 4, 12, 12, 36, 12, 36, 36, 108, 4, 12, 12, 36, 12, 36, 36, 108, 12, 36, 36, 108, 36, 108, 108, 324, 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, 4
Offset: 1

Views

Author

N. J. A. Sloane, Apr 29 2009

Keywords

Comments

Bisection of A323651. - Omar E. Pol, Mar 04 2019

Examples

			From _Omar E. Pol_, Jun 14 2009: (Start)
When written as a triangle:
.1;
.4;
.4,12;
.4,12,12,36;
.4,12,12,36,12,36,36,108;
.4,12,12,36,12,36,36,108,12,36,36,108,36,108,108,324;
.4,12,12,36,12,36,36,108,12,36,36,108,36,108,108,324,12,36,36,108,36,108,...
The rows converge to A161411. (End)

References

  • D. Singmaster, On the cellular automaton of Ulam and Warburton, M500 Magazine of the Open University, #195 (December 2003), pp. 2-7.
  • 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.

Links

Crossrefs

Cf. A147562, A147610 (the sequence divided by 4), A048881, A000120.
Cf. A000079, A161411, A151779, A139250.
Cf. A048883, A139251, A160121, A162349. [Omar E. Pol, Nov 02 2009]
Cf. A323651.

Programs

  • Maple
    A000120 := proc(n) local w,m,i; w := 0; m := n; while m > 0 do i := m mod 2; w := w+i; m := (m-i)/2; od; w; end: wt := A000120; A147582 := n-> if n <= 1 then n else 4*3^(wt(n-1)-1); fi; [seq(A147582(n),n=0..1000)]; # N. J. A. Sloane, Apr 07 2010
  • Mathematica
    s = Plus @@ Flatten@ # & /@ CellularAutomaton[{686, {2, {{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}}, {1, 1}}, {{{1}}, 0}, 200]; f[n_] = If[n == 0, 1, s[[n + 1]] - s[[n]]]; Array[f, 120, 0] (* Michael De Vlieger, Apr 09 2015, after Nadia Heninger and N. J. A. Sloane at A147562 *)

Formula

a(1) = 1; for n > 1, a(n) = 4*3^(wt(n-1)-1) where wt() = A000120(). - R. J. Mathar, Apr 30 2009
This formula is (essentially) given by Singmaster. - N. J. A. Sloane, Aug 06 2009
G.f.: x + 4*x*(Product_{k >= 0} (1 + 3*x^(2^k)) - 1)/3. - N. J. A. Sloane, Jun 10 2009

Extensions

Extended by R. J. Mathar, Apr 30 2009

A160118 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (see Comments for precise definition).

Original entry on oeis.org

0, 1, 9, 13, 41, 45, 73, 85, 169, 173, 201, 213, 297, 309, 393, 429, 681, 685, 713, 725, 809, 821, 905, 941, 1193, 1205, 1289, 1325, 1577, 1613, 1865, 1973, 2729, 2733, 2761, 2773, 2857, 2869, 2953, 2989, 3241, 3253, 3337, 3373, 3625, 3661, 3913, 4021, 4777, 4789
Offset: 0

Views

Author

Omar E. Pol, May 05 2009

Keywords

Comments

On the infinite square grid, we start at stage 0 with all square cells in the OFF state.
Define a "peninsula cell" to a cell that is connected to the structure by exactly one of its vertices.
At stage 1 we turn ON a single cell in the central position.
For n>1, if n is even, at stage n we turn ON all the OFF neighboring cells from cells that were turned in ON at stage n-1.
For n>1, if n is odd, at stage n we turn ON all the peninsular OFF cells.
For the corresponding corner sequence, see A160796.
An animation will show the fractal-like behavior (cf. A139250).
For the first differences see A160415. - Omar E. Pol, Mar 21 2011
First differs from A188343 at a(13). - Omar E. Pol, Mar 28 2011

Examples

			If we label the generations of cells turned ON by consecutive numbers we get the cell pattern shown below:
9...............9
.888.888.888.888.
.878.878.878.878.
.8866688.8866688.
...656.....656...
.8866444.4446688.
.878.434.434.878.
.888.4422244.888.
.......212.......
.888.4422244.888.
.878.434.434.878.
.8866444.4446688.
...656.....656...
.8866688.8866688.
.878.878.878.878.
.888.888.888.888.
9...............9
In the first generation, only the central "1" is ON, a(1)=1. In the next generation, we turn ON eight "2" around the central cell, leading to a(2)=a(1)+8=9. In the third generation, four "3" are turned ON at the vertices of the square, a(3)=a(2)+4=13. And so on...

Links

Crossrefs

Cf. A139250, A139251, A147562, A147610, A160117, A160119, A160379, A160415, A160796, A188343.

Programs

  • Mathematica
    With[{d = 2}, wt[n_] := DigitCount[n, 2, 1]; a[n_] := If[OddQ[n], 3^d + (2^d)*Sum[(2^d - 1)^(wt[k] - 1), {k, 1, (n - 1)/2}] + (2^d)*(3^d - 2)*Sum[(2^d - 1)^(wt[k] - 1), {k, 1, (n - 3)/2}], 3^d + (2^d)*Sum[(2^d - 1)^(wt[k] - 1), {k, 1, n/2 - 1}] + (2^d)*(3^d - 2)*Sum[(2^d - 1)^(wt[k] - 1), {k, 1, n/2 - 1}]]; a[0] = 0; a[1] = 1; Array[a, 50, 0]] (* Amiram Eldar, Aug 01 2023 *)

Formula

From Nathaniel Johnston, Mar 24 2011: (Start)
a(2n-1) = 9 + 4*Sum_{k=2..n} A147610(k) + 28*Sum_{k=2..n-1} A147610(k), n >= 2.
a(2n) = 9 + 4*Sum_{k=2..n} A147610(k) + 28*Sum_{k=2..n} A147610(k), n >= 1.
(End)

Extensions

Entry revised by Omar E. Pol and N. J. A. Sloane, Feb 16 2010, Feb 21 2010
a(8) - a(38) from Nathaniel Johnston, Nov 06 2010
a(13) corrected at the suggestion of Sean A. Irvine. Then I corrected 19 terms between a(14) and a(38). Finally I added a(39)-a(42). - Omar E. Pol, Mar 21 2011
Rule, for n even, edited by Omar E. Pol, Mar 22 2011
Incorrect comment (in "formula" section) removed by Omar E. Pol, Mar 23 2011, with agreement of author.
More terms from Amiram Eldar, Aug 01 2023

A151920 a(n) = (Sum_{i=1..n+1} 3^wt(i))/3, where wt() = A000120().

Original entry on oeis.org

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
Offset: 0

Views

Author

N. J. A. Sloane, Aug 05 2009, Aug 06 2009

Keywords

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

Examples

			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

Links

Crossrefs

Cf. A000120, A130665, A147562, A147610, A153000, A160410, A160414.
Cf. A139250, A151917, A152998.

Programs

  • 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 *)
  • 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

Formula

a(n) = (A147562(n+2) - 1)/4 = (A151917(n+2) - 1)/2. - Omar E. Pol, Mar 13 2011
a(n) = (A130665(n+1) - 1)/3. - Omar E. Pol, Mar 07 2015
a(n) = a(n-1) + 3^A000120(n+1)/3. - David A. Corneth, Mar 21 2015

A151780 a(n) = 5^(wt(n) - 1) where wt(n) = A000120(n).

Original entry on oeis.org

1, 1, 5, 1, 5, 5, 25, 1, 5, 5, 25, 5, 25, 25, 125, 1, 5, 5, 25, 5, 25, 25, 125, 5, 25, 25, 125, 25, 125, 125, 625, 1, 5, 5, 25, 5, 25, 25, 125, 5, 25, 25, 125, 25, 125, 125, 625, 5, 25, 25, 125, 25, 125, 125, 625, 25, 125, 125, 625, 125, 625, 625, 3125, 1, 5, 5, 25, 5, 25, 25, 125, 5
Offset: 1

Views

Author

N. J. A. Sloane, Jun 25 2009

Keywords

Examples

			From _Omar E. Pol_, Jul 21 2009: (Start)
If written as a triangle:
  1;
  1,5;
  1,5,5,25;
  1,5,5,25,5,25,25,125;
  1,5,5,25,5,25,25,125,5,25,25,125,25,125,125,625;
  1,5,5,25,5,25,25,125,5,25,25,125,25,125,125,625,5,25,25,125,25,125,125,625,...
(End)

Crossrefs

Essentially A151779/6.
Cf. A000120, A048881, A048896, A147610, A151783.
Cf. A000351. - Omar E. Pol, Jul 21 2009

Programs

  • PARI
    a(n) = 5^(hammingweight(n)-1); \\ Michel Marcus, Nov 15 2022

A173457 Number of cell turned "ON" at n-th stage of cellular automaton of A173456.

Original entry on oeis.org

0, 1, 8, 12, 4, 28, 36, 4, 28, 36, 12, 84, 108, 4, 28, 36, 12, 84, 108, 12, 84, 108, 36, 252, 324, 4, 28, 36, 12, 84, 108, 12, 84, 108, 36, 252, 324, 12, 84, 108, 36, 252, 324, 36, 252, 324, 108, 756, 972, 4, 28, 36, 12, 84, 108, 12, 84, 108, 36, 252, 324
Offset: 0

Views

Author

Omar E. Pol, Feb 18 2010

Keywords

Comments

Essentially the first differences of A173456.
It appears that row lengths give A098011. After the initial zero, it appears that row lengths give the absolute values of A110164. - Omar E. Pol, Apr 22 2013

Examples

			From Omar E. Pol, Apr 22 2013 (Start):
When written as an irregular triangle begins:
0;
1;
8,12;
4,28,36;
4,28,36,12,84,108;
4,28,36,12,84,108,12,84,108,36,252,324;
4,28,36,12,84,108,12,84,108,36,252,324,12,84,108,36,252,324,36,252,324,108,756,972;
4,28,36,12,84,108,12,84,108,36,252,324,...
(End)

Links

Crossrefs

Cf. A139250, A139251, A173456, A173458, A173459, A173461.

Formula

a(0)=0, a(1)=1, a(2)=8, a(3)=12, for n>=4 when (n MOD 3)=0,1,2 let m=36,4,28 then a(n)=m*A147610((n + 2) / 3). (Found empirically) [Lars Blomberg, Apr 22 2013]

Extensions

a(41)-a(60) from Lars Blomberg, Apr 22 2013

A151783 a(n) = 4^(wt(n) - 1) where wt(n) = A000120(n).

Original entry on oeis.org

1, 1, 4, 1, 4, 4, 16, 1, 4, 4, 16, 4, 16, 16, 64, 1, 4, 4, 16, 4, 16, 16, 64, 4, 16, 16, 64, 16, 64, 64, 256, 1, 4, 4, 16, 4, 16, 16, 64, 4, 16, 16, 64, 16, 64, 64, 256, 4, 16, 16, 64, 16, 64, 64, 256, 16, 64, 64, 256, 64, 256, 256, 1024, 1, 4, 4, 16, 4, 16, 16, 64, 4, 16, 16, 64, 16, 64
Offset: 1

Views

Author

N. J. A. Sloane, Jun 25 2009

Keywords

Examples

			From _Omar E. Pol_, Jul 21 2009: (Start)
If written as a triangle:
  1;
  1,4;
  1,4,4,16;
  1,4,4,16,4,16,16,64;
  1,4,4,16,4,16,16,64,4,16,16,64,16,64,64,256;
  1,4,4,16,4,16,16,64,4,16,16,64,16,64,64,256,4,16,16,64,16,64,64,256,16,64,...
(End)

Crossrefs

Cf. A000120, A048881, A048896, A147610, A151780.
Cf. A000302, A102376. [Omar E. Pol, Jul 21 2009]
A102376 is a very similar sequence.

A173461 Number of cells turned "ON" at n-th stage of cellular automaton of A173460.

Original entry on oeis.org

0, 1, 8, 12, 8, 52, 12, 12, 84, 36, 28, 188, 12, 12, 84, 36, 36, 252, 36, 36, 252, 108, 92, 628, 12, 12, 84, 36, 36, 252, 36, 36, 252, 108, 108, 756, 36, 36, 252, 108, 108, 756, 108, 108, 756, 324, 292, 2012, 12, 12, 84, 36, 36, 252, 36, 36, 252, 108, 108
Offset: 0

Views

Author

Omar E. Pol, Feb 18 2010

Keywords

Comments

Essentially the first differences of A173460.
It appears that row lengths give the absolute values of A110164. - Omar E. Pol, Apr 25 2013

Examples

			From _Omar E. Pol_, Apr 25 2013: (Start)
When written as an irregular triangle begins:
0;
1,8;
12,8,52;
12,12,84,36,28,188;
12,12,84,36,36,252,36,36,252,108,92,628;
12,12,84,36,36,252,36,36,252,108,108,756,36,36,252,108,108,756,108,108,756,324,292,2012;
12,12,84,36,36,252,36,36,252,108,108,...
(End)

Links

Crossrefs

Cf. A139250, A139251, A173457, A173460, A173462, A173463.

Formula

a(0)=0, a(1)=1, a(2)=8, for n>=3 let i=n/3+1, j=A147610(i), if 2^r==i for some r then let c1=2^(r+1), c2=2^(r+4) else let c1=c2=0, finally when (n MOD 3)=0,1,2 let a(n)=12*j, 12*j-c1, 84*j-c2. (Found empirically) [Lars Blomberg, Apr 23 2013]

Extensions

More terms a(14)-a(17) from Omar E. Pol, Sep 25 2011
a(18)-a(58) from Lars Blomberg, Apr 23 2013

A183060 Number of "ON" cells at n-th stage in a simple 2-dimensional cellular automaton (see Comments for precise definition).

Original entry on oeis.org

0, 1, 4, 7, 14, 17, 24, 31, 50, 53, 60, 67, 86, 93, 112, 131, 186, 189, 196, 203, 222, 229, 248, 267, 322, 329, 348, 367, 422, 441, 496, 551, 714, 717, 724, 731, 750, 757, 776, 795, 850, 857, 876, 895, 950, 969, 1024, 1079, 1242, 1249, 1268, 1287
Offset: 0

Views

Author

Omar E. Pol, Feb 20 2011

Keywords

Comments

On the semi-infinite square grid, start with all cells OFF.
Turn a single cell to the ON state in row 1.
At each subsequent step, each cell with exactly one neighbor ON is turned ON, and everything that is already ON remains ON.
The sequence gives the number of "ON" cells after n stages. A183061 (the first differences) gives the number of cells turned "ON" at the n-th stage.
Note that this is just half plus the rest of the center line of the cellular automaton described in A147562.
After 2^k stages the structure resembles an isosceles right triangle. For a three-dimensional version using cubes see A186410. For more information see A147562.

Examples

			Illustration of the structure after eight stages in which we label the generations of cells turned ON by consecutive numbers:
         8
        878
       8 6 8
      8765678
     8 8 4 8 8
    878 434 878
   8 6 4 2 4 6 8
  876543212345678
...................
There are 50 "ON" cells so a(8) = 50.

Links

Crossrefs

Cf. A139250, A147562, A147582, A147610, A151920, A183061, A186410.

Programs

  • Mathematica
    A183060[0] = 0; A183060[n_] := Total[With[{m = n - 1}, CellularAutomaton[{4042387958, 2, {{0, 1}, {-1, 0}, {0, 0}, {1, 0}, {0, -1}}}, {{{1}}, 0}, {{{m}}, -m}]], 2] (* JungHwan Min, Jan 24 2016 *)
A183060[0] = 0; A183060[n_] := Total[With[{m = n - 1}, CellularAutomaton[{686, {2, {{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}}, {1, 1}}, {{{1}}, 0}, {{{m}}, -m}]], 2] (* JungHwan Min, Jan 24 2016 *)

Formula

a(n) = n + (A147562(n) - 1)/2, n >= 1.
a(n) = n + 2*A151920(n-2), n >= 2.
a(2^n) = A076024(n+1). - Nathaniel Johnston, Mar 14 2011

Extensions

Comments edited by Omar E. Pol, Mar 19 2011 at the suggestion of John W. Layman and Franklin T. Adams-Watters
Showing 1-10 of 14 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.