A071053 -id:A071053 - cp's OEIS frontend

A246674 Run Length Transform of A000225.

1, 1, 1, 3, 1, 1, 3, 7, 1, 1, 1, 3, 3, 3, 7, 15, 1, 1, 1, 3, 1, 1, 3, 7, 3, 3, 3, 9, 7, 7, 15, 31, 1, 1, 1, 3, 1, 1, 3, 7, 1, 1, 1, 3, 3, 3, 7, 15, 3, 3, 3, 9, 3, 3, 9, 21, 7, 7, 7, 21, 15, 15, 31, 63, 1, 1, 1, 3, 1, 1, 3, 7, 1, 1, 1, 3, 3, 3, 7, 15, 1, 1, 1, 3, 1, 1, 3, 7, 3, 3, 3, 9, 7, 7, 15, 31, 3, 3, 3, 9, 3, 3, 9, 21, 3, 3, 3, 9, 9, 9, 21, 45, 7, 7, 7, 21, 7, 7, 21, 49, 15, 15, 15, 45, 31, 31, 63, 127, 1

Offset: 0

Antti Karttunen, Sep 08 2014

a(n) can be also computed by replacing all consecutive runs of zeros in the binary expansion of n with * (multiplication sign), and then performing that multiplication, still in binary, after which the result is converted into decimal. See the example below.

			115 is '1110011' in binary. The run lengths of 1-runs are 2 and 3, thus a(115) = A000225(2) * A000225(3) = ((2^2)-1) * ((2^3)-1) = 3*7 = 21.
The same result can be also obtained more directly, by realizing that '111' and '11' are the binary representations of 7 and 3, and 7*3 = 21.
From _Omar E. Pol_, Feb 15 2015: (Start)
Written as an irregular triangle in which row lengths are the terms of A011782:
1;
1;
1,3;
1,1,3,7;
1,1,1,3,3,3,7,15;
1,1,1,3,1,1,3,7,3,3,3,9,7,7,15,31;
1,1,1,3,1,1,3,7,1,1,1,3,3,3,7,15,3,3,3,9,3,3,9,21,7,7,7,21,15,15,31,63;
...
Right border gives 1 together with the positive terms of A000225.
(End)

Antti Karttunen, Table of n, a(n) for n = 0..8192

Cf. A003714 (gives the positions of ones).
Cf. A000225, A051179.
A001316 is obtained when the same transformation is applied to A000079, the powers of two.
Run Length Transforms of other sequences: A071053, A227349, A246588, A246595, A246596, A246660, A246661, A246685, A247282.

f[n_] := 2^n - 1; Table[Times @@ (f[Length[#]]&) /@ Select[ Split[ IntegerDigits[n, 2]], #[[1]] == 1&], {n, 0, 100}] (* Jean-François Alcover, Jul 11 2017 *)

# uses RLT function in A278159
def A246674(n): return RLT(n,lambda m: 2**m-1) # Chai Wah Wu, Feb 04 2022

For all n >= 0, a(A051179(n)) = A247282(A051179(n)) = A051179(n).

A247649 Number of terms in expansion of f^n mod 2, where f = 1/x^2 + 1/x + 1 + x + x^2 mod 2.

Original entry on oeis.org

1, 5, 5, 7, 5, 17, 7, 19, 5, 25, 17, 19, 7, 31, 19, 25, 5, 25, 25, 35, 17, 61, 19, 71, 7, 35, 31, 41, 19, 71, 25, 77, 5, 25, 25, 35, 25, 85, 35, 95, 17, 85, 61, 71, 19, 91, 71, 77, 7, 35, 35, 49, 31, 107, 41, 121, 19, 95, 71, 85, 25, 113, 77, 103

Offset: 0

N. J. A. Sloane, Sep 25 2014

nonn

This is the number of cells that are ON after n generations in a one-dimensional cellular automaton defined by the odd-neighbor rule where the neighborhood consists of 5 contiguous cells.

a(n) is also the number of odd entries in row n of A035343. - Leon Rische, Feb 02 2023

			The first few generations are:
..........X..........
........XXXXX........
......X.X.X.X.X......
....XX..X.X.X..XX.... (f^3)
..X...X...X...X...X..
XXXX.XXX.XXX.XXX.XXXX
...
f^3 mod 2 = x^6 + x^5 + x^2 + 1/x^2 + 1/x^5 + 1/x^6 + 1 has 7 terms, so a(3) = 7.
From _Omar E. Pol_, Mar 02 2015: (Start)
Also, written as an irregular triangle in which the row lengths are the terms of A011782, the sequence begins:
  1;
  5;
  5, 7;
  5,17, 7,19;
  5,25,17,19, 7,31,19,25;
  5,25,25,35,17,61,19,71, 7,35,31,41,19,71,25,77;
  5,25,25,35,25,85,35,95,17,85,61,71,19,91,71,77,7,35,35,49,31,107,41,121,19, ...
(End)
It follows from the Generalized Run Length Transform result mentioned in the comments that in each row the first quarter of the terms (and no more) are equal to 5 times the beginning of the sequence itself. It cannot be said that the rows converge (in any meaningful sense) to five times the sequence. - _N. J. A. Sloane_, Mar 03 2015

Chai Wah Wu, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Index entries for sequences related to cellular automata

Cf. A071053, A247647, A247648, A253085, A255490.

Partial sums are in A255654.

import sympy
from functools import reduce
from operator import mul
x = sympy.symbols('x')
f = 1/x**2+1/x+1+x+x**2
A247649_list, g = [1], 1
for n in range(1,1001):
    s = [int(d,2) for d in bin(n)[2:].split('00') if d != '']
    g = (g*f).expand(modulus=2)
    if len(s) == 1:
        A247649_list.append(g.subs(x,1))
    else:
        A247649_list.append(reduce(mul,(A247649_list[d] for d in s)))
# Chai Wah Wu, Sep 25 2014

The values of a(n) for n in A247647 (or A247648) determine all the values, as follows. Parse the binary expansion of n into terms from A247647 separated by at least two zeros: m_1 0...0 m_2 0...0 m_3 ... m_r 0...0. Ignore any number (one or more) of trailing zeros. Then a(n) = a(m_1)*a(m_2)*...*a(m_r). For example, n = 37_10 = 100101_2 is parsed into 1.00.101, and so a(37) = a(1)*a(5) = 5*17 = 85. This is a generalization of the Run Length Transform.

A246034 Number of odd terms in f^n, where f = x^4y^4 + x^4y^3 + x^3y^4 + x^4y^2 + x^2y^4 + x^4y + x^3y^2 + x^2y^3 + xy^4 + x^4 + x^2y^2 + y^4 + x^3 + x^2y + xy^2 + y^3 + x^2 + y^2 + x + y + 1.

Original entry on oeis.org

1, 21, 21, 85, 21, 233, 85, 321, 21, 441, 233, 761, 85, 1137, 321, 1545, 21, 441, 441, 1785, 233, 2925, 761, 3589, 85, 1785, 1137, 3977, 321, 4549, 1545, 5909, 21, 441, 441, 1785, 441, 4893, 1785, 6741, 233, 4893, 2925, 9949, 761, 11301, 3589, 13181, 85, 1785, 1785

Offset: 0

N. J. A. Sloane, Aug 20 2014

nonn

This is the number of ON cells in a certain 2-D CA in which the neighborhood of a cell is defined by f, and in which a cell is ON iff there was an odd number of ON cells in the neighborhood at the previous generation.

			Here is the neighborhood:
[X, X, X, X, X]
[X, 0, X, 0, X]
[X, X, X, X, X]
[X, 0, X, 0, X]
[X, X, X, X, X]
which contains a(1) = 21 ON cells.

Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata, arXiv:1503.01796 [math.CO], 2015; see also the Accompanying Maple Package.
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, Odd-Rule Cellular Automata on the Square Grid, arXiv:1503.04249 [math.CO], 2015.
N. J. A. Sloane, On the No. of ON Cells in Cellular Automata, Video of talk in Doron Zeilberger's Experimental Math Seminar at Rutgers University, Feb. 05 2015: Part 1, Part 2
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Index entries for sequences related to cellular automata

Other CA's that use the same rule but with different cell neighborhoods: A160239, A102376, A071053, A072272, A001316, A246035.

C:=f->subs({x=1, y=1}, f);
# Find number of ON cells in CA for generations 0 thru M defined by rule
# that cell is ON iff number of ON cells in nbd at time n-1 was odd
# where nbd is defined by a polynomial or Laurent series f(x, y).
OddCA:=proc(f, M) global C; local n, a, i, f2, p;
f2:=simplify(expand(f)) mod 2;
a:=[]; p:=1;
for n from 0 to M do a:=[op(a), C(p)]; p:=expand(p*f2) mod 2; od:
lprint([seq(a[i], i=1..nops(a))]);
end;
f:=x^4*y^4+x^4*y^3+x^3*y^4+x^4*y^2+x^2*y^4+x^4*y+x^3*y^2+x^2*y^3+x*y^4+x^4+
   x^2*y^2+y^4+x^3+x^2*y+x*y^2+y^3+x^2+y^2+x+y+1;
OddCA(f, 100);

f = x^4*y^4 + x^4*y^3 + x^3*y^4 + x^4*y^2 + x^2*y^4 + x^4*y + x^3*y^2 + x^2*y^3 + x*y^4 + x^4 + x^2*y^2 + y^4 + x^3 + x^2*y + x*y^2 + y^3 + x^2 + y^2 + x + y + 1;
a[0] = 1; a[n_] := Count[List @@ Expand[f^n] /. {x -> 1, y -> 1}, _?OddQ];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Dec 11 2017 *)

A071036 Triangle read by rows giving successive states of cellular automaton generated by "Rule 150" when started with a single ON cell.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1

Offset: 0

Hans Havermann, May 26 2002

Row n has length 2n+1.
Also the coefficients of (x^2 + x + 1)^n mod 2. - Alan DenAdel, Mar 19 2014
The number of 0's in row n is A071052(n), and the number of 1's in row n is A071053(n). - Michael Somos, Jun 24 2018

			Triangle begins:
               1;
            1, 1, 1;
         1, 0, 1, 0, 1;
      1, 1, 0, 1, 0, 1, 1;
   1, 0, 0, 0, 1, 0, 0, 0, 1;
1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1;
... - _Michel Marcus_, Mar 20 2014

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; Chapter 3.

Robert Price, Table of n, a(n) for n = 0..9999
Rémy Sigrist, Representation of the first 2^10 rows of the table
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index to Elementary Cellular Automata
Index entries for sequences related to cellular automata

Cf. A027907, A071052, A071053.

This sequence, A038184 and A118110 are equivalent descriptions of the Rule 150 automaton.

T[ n_, k_] := T[n, k] = Which[k < 0 || k > 2 n, 0, n == k == 0, 1, True, Mod[ T[n - 1, k - 2] + T[n - 1, k - 1] + T[n - 1, k], 2]]; (* Michael Somos, Jun 24 2018 *)

rown(n) = Vec(lift((x^2 + x + 1)^n * Mod(1, 2))); \\ Michel Marcus, Mar 20 2014

a(n) = A027907(n) modulo 2. - Michel Marcus, Mar 20 2014

Corrected by Hans Havermann, Jan 08 2012

A134659 Total number of odd coefficients in (1+x+x^2)^k for k=0,...,n.

Original entry on oeis.org

1, 4, 7, 12, 15, 24, 29, 40, 43, 52, 61, 76, 81, 96, 107, 128, 131, 140, 149, 164, 173, 200, 215, 248, 253, 268, 283, 308, 319, 352, 373, 416, 419, 428, 437, 452, 461, 488, 503, 536, 545, 572, 599, 644, 659, 704, 737, 800, 805, 820, 835, 860, 875, 920, 945, 1000

Offset: 0

Steven Finch, Jan 25 2008

nonn

a(n) = Sum_{k <= n} A071053(k)

N. J. A. Sloane, Table of n, a(n) for n = 0..16383
S. R. Finch, P. Sebah and Z.-Q. Bai, Odd Entries in Pascal's Trinomial Triangle (arXiv:0802.2654)
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168, 2015

Cf. A001316, A006046, A027907, A071053.

Sum[PolynomialMod[(1+x+x^2)^k, 2] /. x->1, {k, 0, n-1}]

Offset changed to 0 by N. J. A. Sloane, Feb 06 2015

A245564 a(n) = Product_{i in row n of A245562} Fibonacci(i+2).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 3, 5, 2, 4, 4, 6, 3, 6, 5, 8, 2, 4, 4, 6, 4, 8, 6, 10, 3, 6, 6, 9, 5, 10, 8, 13, 2, 4, 4, 6, 4, 8, 6, 10, 4, 8, 8, 12, 6, 12, 10, 16, 3, 6, 6, 9, 6, 12, 9, 15, 5, 10, 10, 15, 8, 16, 13, 21, 2, 4, 4, 6, 4, 8, 6, 10, 4, 8, 8, 12, 6, 12, 10, 16, 4, 8, 8, 12, 8, 16, 12, 20, 6, 12, 12, 18

Offset: 0

N. J. A. Sloane, Aug 10 2014; revised Sep 05 2014

This is the Run Length Transform of S(n) = Fibonacci(n+2).
The Run Length Transform of a sequence {S(n), n>=0} is defined to be the sequence {T(n), n>=0} given by T(n) = Product_i S(i), where i runs through the lengths of runs of 1's in the binary expansion of n. E.g. 19 is 10011 in binary, which has two runs of 1's, of lengths 1 and 2. So T(19) = S(1)*S(2). T(0)=1 (the empty product).
Also the number of sparse subsets of the binary indices of n, where a set is sparse iff 1 is not a first difference. The maximal case is A384883. For prime instead of binary indices we have A166469. - Gus Wiseman, Jul 05 2025

			From _Gus Wiseman_, Jul 05 2025: (Start)
The binary indices of 11 are {1,2,4}, with sparse subsets {{},{1},{2},{4},{1,4},{2,4}}, so a(11) = 6.
The maximal runs of binary indices of 11 are ((1,2),(4)), with lengths (2,1), so a(11) = F(2+2)*F(1+2) = 6.
The a(0) = 1 through a(12) = 3 sparse subsets are:
  0    1    2    3    4    5    6    7    8    9    10    11    12
  ------------------------------------------------------------------
  {}   {}   {}   {}   {}   {}   {}   {}   {}   {}    {}    {}    {}
       {1}  {2}  {1}  {3}  {1}  {2}  {1}  {4}  {1}   {2}   {1}   {3}
                 {2}       {3}  {3}  {2}       {4}   {4}   {2}   {4}
                           {1,3}     {3}       {1,4} {2,4} {4}
                                     {1,3}                 {1,4}
                                                           {2,4}
The greatest number whose set of binary indices is a member of column n above is A374356(n).
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..8191
Chai Wah Wu, Sums of products of binomial coefficients mod 2 and run length transforms of sequences, arXiv:1610.06166 [math.CO], 2016.

Cf. A245562, A000045, A001045, A071053, A245565, A246028.
A034839 counts subsets by number of maximal runs, strict partitions A116674.
A384877 gives lengths of maximal anti-runs of binary indices, firsts A384878.
A384893 counts subsets by number of maximal anti-runs, for partitions A268193, A384905.
Cf. A000071, A001629, A010049, A053538, A166469, A202023, A202064, A208342, A374356, A384883.

with(combinat); ans:=[];
for n from 0 to 100 do lis:=[]; t1:=convert(n,base,2); L1:=nops(t1); out1:=1; c:=0;
for i from 1 to L1 do
   if out1 = 1 and t1[i] = 1 then out1:=0; c:=c+1;
   elif out1 = 0 and t1[i] = 1 then c:=c+1;
   elif out1 = 1 and t1[i] = 0 then c:=c;
   elif out1 = 0 and t1[i] = 0 then lis:=[c,op(lis)]; out1:=1; c:=0;
   fi;
   if i = L1 and c>0 then lis:=[c,op(lis)]; fi;
                   od:
a:=mul(fibonacci(i+2), i in lis);
ans:=[op(ans),a];
od:
ans;

a[n_] := Sum[Mod[Binomial[3k, k] Binomial[n, k], 2], {k, 0, n}];
a /@ Range[0, 100] (* Jean-François Alcover, Feb 29 2020, after Chai Wah Wu *)
spars[S_]:=Select[Subsets[S],FreeQ[Differences[#],1]&];
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Length[spars[bpe[n]]],{n,0,30}] (* Gus Wiseman, Jul 05 2025 *)

a(n)=my(s=1,k); while(n, n>>=valuation(n,2); k=valuation(n+1,2); s*=fibonacci(k+2); n>>=k); s \\ Charles R Greathouse IV, Oct 21 2016

# use RLT function from A278159
from sympy import fibonacci
def A245564(n): return RLT(n,lambda m: fibonacci(m+2)) # Chai Wah Wu, Feb 04 2022

a(n) = Sum_{k=0..n} ({binomial(3k,k)*binomial(n,k)} mod 2). - Chai Wah Wu, Oct 19 2016

A247640 Number of ON cells after n generations of "Odd-Rule" cellular automaton on hexagonal lattice based on 6-celled neighborhood.

Original entry on oeis.org

1, 6, 6, 24, 6, 36, 24, 96, 6, 36, 36, 144, 24, 144, 96, 384, 6, 36, 36, 144, 36, 216, 144, 576, 24, 144, 144, 576, 96, 576, 384, 1536, 6, 36, 36, 144, 36, 216, 144, 576, 36, 216, 216, 864, 144, 864, 576, 2304, 24, 144, 144, 576, 144, 864

Offset: 0

N. J. A. Sloane, Sep 22 2014

nonn

The neighborhood of a cell consists of the six surrounding cells (but not the cell itself). A cell is ON at generation n iff an odd number of its neighbors were ON at the previous generation. We start with one ON cell.
This is the Run Length Transform of the sequence 1, 6, 24, 96, 384, 1536, 6144, 24576, ... (almost certainly A164908, or 1 followed by A002023).
It appears that this is also the sequence corresponding to the odd-rule cellular automaton defined by OddRule 356 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link). - N. J. A. Sloane, Feb 26 2015

Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata, arXiv:1503.01796 [math.CO], 2015; see also the Accompanying Maple Package.
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, Odd-Rule Cellular Automata on the Square Grid, arXiv:1503.04249 [math.CO], 2015.
N. J. A. Sloane, On the No. of ON Cells in Cellular Automata, Video of talk in Doron Zeilberger's Experimental Math Seminar at Rutgers University, Feb. 05 2015: Part 1, Part 2
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Index entries for sequences related to cellular automata

Cf. A164908, A001023, A071053, A160239, A247666.

C := f->`if`(type(f,`+`),nops(f),1);
f := 1/x+x+1/y+y+1/(x*y)+x*y;
g := n->expand(f^n) mod 2;
[seq(C(g(n)),n=0..100)];

A247640[n_] := Total[CellularAutomaton[{42, {2, {{1, 1, 0}, {1, 0, 1}, {0, 1, 1}}}, {1, 1}}, {{{1}}, 0}, {{{n}}}], 2]; Array[A247640, 54, 0] (* JungHwan Min, Sep 06 2016 *)
A247640L[n_] := Total[#, 2] & /@ CellularAutomaton[{42, {2, {{1, 1, 0}, {1, 0, 1}, {0, 1, 1}}}, {1, 1}}, {{{1}}, 0}, n]; A247640L[53] (* JungHwan Min, Sep 06 2016 *)

a(n) = number of terms in expansion of f^n mod 2, where f = 1/x+x+1/y+y+1/(x*y)+x*y (mod 2);

A247666 Number of ON cells after n generations of "Odd-Rule" cellular automaton on hexagonal lattice based on 7-celled neighborhood.

Original entry on oeis.org

1, 7, 7, 25, 7, 49, 25, 103, 7, 49, 49, 175, 25, 175, 103, 409, 7, 49, 49, 175, 49, 343, 175, 721, 25, 175, 175, 625, 103, 721, 409, 1639, 7, 49, 49, 175, 49, 343, 175, 721, 49, 343, 343, 1225, 175, 1225, 721, 2863, 25, 175, 175, 625

Offset: 0

N. J. A. Sloane, Sep 22 2014

nonn

The neighborhood of a cell consists of the cell itself together with its six surrounding cells. A cell is ON at generation n iff an odd number of its neighbors were ON at the previous generation. We start with one ON cell.
This is the Run Length Transform of the sequence 1,7,25,103,409,1639,26215,... (almost certainly A102900).
This appears to be the same as the number of ON cells in a certain 2-D CA on the square grid in which the neighborhood of a cell is defined by f = 1/(x*y)+1/x+1/x*y+1/y+x/y+x+x*y, and in which a cell is ON iff there was an odd number of ON cells in the neighborhood at the previous generation. Here is the neighborhood:
[X, 0, X]
[X, 0, X]
[X, X, X]
which contains a(1) = 7 ON cells.
This is the odd-rule cellular automaton defined by OddRule 557 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link).
Furthermore, this is also the number of ON cells in the 2-D CA on the square grid in which the neighborhood of a cell is defined by f = 1/(x*y)+1/x+1/y+1+y+x+x*y, with the same rule. Here is the neighborhood:
[0, X, X]
[X, X, X]
[X, X, 0]
- N. J. A. Sloane, Feb 19 2015
This is the odd-rule cellular automaton defined by OddRule 376 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link).
The partial sums are in A253767 in which the structure looks like an irregular stepped pyramid, apparently with a like-hexagonal base. - Omar E. Pol, Jan 29 2015

			From _Omar E. Pol_, Jan 29 2015: (Start)
May be arranged into blocks of sizes A011782:
1;
7;
7, 25;
7, 49, 25, 103;
7, 49, 49, 175, 25, 175, 103, 409;
7, 49, 49, 175, 49, 343, 175, 721, 25, 175, 175, 625, 103, 721, 409, 1639;
7, 49, 49, 175, 49, 343, 175, 721, 49, 343, 343, 1225, 175, 1225, 721, 2863, 25, 175, 175, 625, ...
It appears that right border gives A102900 without repetitions, see Comments section. [This is just a restatement of the fact that this sequence is the run length transform of what is presumably A102900. - _N. J. A. Sloane_, Feb 06 2015]
(End)
From _Omar E. Pol_, Mar 19 2015: (Start)
Also, the sequence can be written as an irregular tetrahedron T(s,r,k) as shown below:
1;
..
7;
..
7;
25;
.........
7,    49;
25;
103;
...................
7,    49,  49, 175;
25,  175;
103;
409;
......................................
7,    49,  49, 175, 49, 343, 175, 721;
25,  175, 175, 625;
103, 721;
409;
1639;
...
Apart from the initial 1, we have that T(s,r,k) = T(s+1,r,k).
(End)

Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata, arXiv:1503.01796 [math.CO], 2015; see also the Accompanying Maple Package.
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, Odd-Rule Cellular Automata on the Square Grid, arXiv:1503.04249 [math.CO], 2015.
N. J. A. Sloane, Illustrations of generations 0 to 4
N. J. A. Sloane, On the No. of ON Cells in Cellular Automata, Video of talk in Doron Zeilberger's Experimental Math Seminar at Rutgers University, Feb. 05 2015: Part 1, Part 2
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Index entries for sequences related to cellular automata

Cf. A102900, A071053, A160239, A247640.

C := f->`if`(type(f,`+`),nops(f),1);
f := 1+1/x+x+1/y+y+1/(x*y)+x*y;
g := n->expand(f^n) mod 2;
[seq(C(g(n)),n=0..100)];

A247666[n_] := Total[CellularAutomaton[{170, {2, {{1, 1, 0}, {1, 1, 1}, {0, 1, 1}}}, {1, 1}}, {{{1}}, 0}, {{{n}}}], 2]; Array[A247666, 52, 0] (* JungHwan Min, Sep 01 2016 *)
A247666L[n_] := Total[#, 2] & /@ CellularAutomaton[{170, {2, {{1, 1, 0}, {1, 1, 1}, {0, 1, 1}}}, {1, 1}}, {{{1}}, 0}, n]; A247666L[51] (* JungHwan Min, Sep 01 2016 *)

a(n) = number of terms in expansion of f^n mod 2, where f = 1+1/x+x+1/y+y+1/(x*y)+x*y (mod 2);

A255488 Number of odd terms in expansion of (1 + x + x^2 + x^3 + x^4 + x^5)^n.

Original entry on oeis.org

1, 6, 6, 12, 6, 16, 12, 24, 6, 36, 16, 32, 12, 36, 24, 48, 6, 36, 36, 72, 16, 56, 32, 64, 12, 72, 36, 72, 24, 68, 48, 96, 6, 36, 36, 72, 36, 96, 72, 144, 16, 96, 56, 112, 32, 100, 64, 128, 12, 72, 72, 144, 36, 120, 72, 144, 24, 144, 68, 136

Offset: 0

N. J. A. Sloane, Mar 01 2015

nonn

All the following are of the same type: A001316, A071053, A134660, A134661, A134662, A255485, A247649, A255486. It would be nice to have some unifying formula or recurrence. (Restating the definition, these are the Hamming weights of the n-th powers of the corresponding polynomials over GF(2). - Joerg Arndt, Mar 02 2015)

			From _Omar E. Pol_, Mar 01 2015: (Start)
Written as an irregular triangle in which the row lengths are the terms of A011782, the sequence begins:
1;
6;
6,12;
6,16,12,24;
6,36,16,32,12,36,24,48;
6,36,36,72,16,56,32,64,12,72,36,72,24,68,48,96;
6,36,36,72,36,96,72,144,16,96,56,112,32,100,64,128,12,72,72,144,36,120,72,144,24,144,68,136...
...
In each row the first quarter of the terms (and no more) are equal to 6 times the beginning of the sequence itself (corrected after Sloane's comment in A247649, Mar 03 2015).
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..8191

Cf. A001316, A071053, A134660, A134661, A134662, A255485, A247649, A255486.

r1:=proc(f) local g,n; g:=n->nops(expand(f^n) mod 2); [seq(g(n),n=0..90)]; end;
r1(1+x+x^2+x^3);

a[n_] := Count[CoefficientList[(1 + x + x^2 + x^3 + x^4 + x^5)^n, x], _?OddQ];
Table[a[n], {n, 0, 90}] (* Jean-François Alcover, Apr 06 2017 *)

a(n) = {my(pol=(1+x+x^2+x^3+x^4+x^5)*Mod(1,2)); subst(lift(pol^n), x, 1);} \\ Michel Marcus, Mar 01 2015

A253064 Number of odd terms in f^n, where f = 1/x + 1 + x + y.

Original entry on oeis.org

1, 4, 4, 12, 4, 16, 12, 40, 4, 16, 16, 48, 12, 48, 40, 128, 4, 16, 16, 48, 16, 64, 48, 160, 12, 48, 48, 144, 40, 160, 128, 416, 4, 16, 16, 48, 16, 64, 48, 160, 16, 64, 64, 192, 48, 192, 160, 512, 12, 48, 48, 144, 48, 192, 144, 480, 40, 160, 160, 480, 128, 512, 416

Offset: 0

N. J. A. Sloane, Jan 26 2015

nonn

This is the number of ON cells in a certain 2-D CA in which the neighborhood of a cell is defined by f, and in which a cell is ON iff there was an odd number of ON cells in the neighborhood at the previous generation.

This is the odd-rule cellular automaton defined by OddRule 017 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link). - N. J. A. Sloane, Feb 25 2015

			Here is the neighborhood f:
[0, X, 0]
[X, X, X]
which contains a(1) = 4 ON cells.

Alois P. Heinz, Table of n, a(n) for n = 0..8191
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata, arXiv:1503.01796, 2015; see also the Accompanying Maple Package.
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, Odd-Rule Cellular Automata on the Square Grid, arXiv:1503.04249, 2015.
Po-Yi Huang and Wen-Fong Ke, Sequences Derived from The Symmetric Powers of {1,2,...,k}, arXiv:2307.07733 [math.CO], 2023.
N. J. A. Sloane, On the No. of ON Cells in Cellular Automata, Video of talk in Doron Zeilberger's Experimental Math Seminar at Rutgers University, Feb. 05 2015: Part 1, Part 2
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168, 2015
Index entries for sequences related to cellular automata

Other CA's that use the same rule but with different cell neighborhoods: A160239, A102376, A071053, A072272, A001316, A246034, A246035.

Cf. A087206.

C:=f->subs({x=1, y=1}, f);
# Find number of ON cells in CA for generations 0 thru M defined by rule
# that cell is ON iff number of ON cells in nbd at time n-1 was odd
# where nbd is defined by a polynomial or Laurent series f(x, y).
OddCA:=proc(f, M) global C; local n, a, i, f2, p;
f2:=simplify(expand(f)) mod 2;
a:=[]; p:=1;
for n from 0 to M do a:=[op(a), C(p)]; p:=expand(p*f2) mod 2; od:
lprint([seq(a[i], i=1..nops(a))]);
end;
f:=1/x+1+x+y;
OddCA(f, 130);

f[n_] := 2^n*Fibonacci[n+2]; Table[Times @@ (f[Length[#]]&) /@ Select[ Split[ IntegerDigits[n, 2]], #[[1]] == 1&], {n, 0, 62}] (* Jean-François Alcover, Jul 11 2017 *)

This is the Run Length Transform of A087206.

cp's OEIS Frontend

A246674 Run Length Transform of A000225.

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A247649 Number of terms in expansion of f^n mod 2, where f = 1/x^2 + 1/x + 1 + x + x^2 mod 2.

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A246034 Number of odd terms in f^n, where f = x^4*y^4 + x^4*y^3 + x^3*y^4 + x^4*y^2 + x^2*y^4 + x^4*y + x^3*y^2 + x^2*y^3 + x*y^4 + x^4 + x^2*y^2 + y^4 + x^3 + x^2*y + x*y^2 + y^3 + x^2 + y^2 + x + y + 1.

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A071036 Triangle read by rows giving successive states of cellular automaton generated by "Rule 150" when started with a single ON cell.

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A134659 Total number of odd coefficients in (1+x+x^2)^k for k=0,...,n.

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A245564 a(n) = Product_{i in row n of A245562} Fibonacci(i+2).

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A247640 Number of ON cells after n generations of "Odd-Rule" cellular automaton on hexagonal lattice based on 6-celled neighborhood.

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A246034 Number of odd terms in f^n, where f = x^4y^4 + x^4y^3 + x^3y^4 + x^4y^2 + x^2y^4 + x^4y + x^3y^2 + x^2y^3 + xy^4 + x^4 + x^2y^2 + y^4 + x^3 + x^2y + xy^2 + y^3 + x^2 + y^2 + x + y + 1.