A071053 -id:A071053 - cp's OEIS frontend

A253102 a(n) = A071053(n)^3.

1, 27, 27, 125, 27, 729, 125, 1331, 27, 729, 729, 3375, 125, 3375, 1331, 9261, 27, 729, 729, 3375, 729, 19683, 3375, 35937, 125, 3375, 3375, 15625, 1331, 35937, 9261, 79507, 27, 729, 729, 3375, 729, 19683, 3375, 35937, 729, 19683, 19683, 91125, 3375, 91125, 35937, 250047, 125, 3375, 3375, 15625

Offset: 0

N. J. A. Sloane, Feb 20 2015

nonn

Number of ON cells at n-th generation of 3-D CA defined by generalization of Rule 150, starting with a single ON cell at generation 0.
Number of odd coefficients in ((1/x+1+x)*(1/y+1+y)*(1/z+1+z))^n.
Run length transform of A253103.

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. A071053, A246035, A253103.

a71053[n_] := Total[CoefficientList[(x^2 + x + 1)^n, x, Modulus -> 2]];
Table[a71053[n]^3, {n, 0, 51}] (* Jean-François Alcover, Sep 15 2018 *)

A245562 Table read by rows: row n gives list of lengths of runs of 1's in binary expansion of n, starting with high-order bits.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 3, 4, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 3, 2, 2, 1, 2, 1, 2, 2, 3, 3, 1, 4, 5, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 3, 1, 4, 2, 2, 1

Offset: 0

N. J. A. Sloane, Aug 10 2014

A formula for A071053(n) depends on this table.

			Here are the run lengths for the numbers 0 through 21:
0, []
1, [1]
2, [1]
3, [2]
4, [1]
5, [1, 1]
6, [2]
7, [3]
8, [1]
9, [1, 1]
10, [1, 1]
11, [1, 2]
12, [2]
13, [2, 1]
14, [3]
15, [4]
16, [1]
17, [1, 1]
18, [1, 1]
19, [1, 2]
20, [1, 1]
21, [1, 1, 1]

Reinhard Zumkeller, Table of n, a(n) for n = 0..2770
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168, 2015

Row sums = A000120 (the binary weight).

Cf. A245563, A071053.

for n from 0 to 128 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:
lprint(n,lis);
od:

row(n)=my(v=List(),k); while(n, n>>=valuation(n,2); listput(v, k=valuation(n+1,2)); n>>=k); Vecrev(v) \\ Charles R Greathouse IV, Oct 21 2016

from re import split
A245562_list = [0]
for n in range(1,100):
    A245562_list.extend(len(d) for d in split('0+',bin(n)[2:]) if d != '')
# Chai Wah Wu, Sep 07 2014

A245563 Table read by rows: row n gives list of lengths of runs of 1's in binary expansion of n, starting with low-order bits.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 3, 4, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 2, 2, 3, 1, 3, 4, 5, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 3, 1, 4, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 1, 2, 2, 2, 3, 2, 3, 1, 3, 1, 3, 2, 3, 4, 1, 4, 5, 6, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 1, 1

Offset: 0

N. J. A. Sloane, Aug 10 2014

A formula for A071053(n) depends on this table.

			Here are the run lengths for the numbers 0 through 21:
0, []
1, [1]
2, [1]
3, [2]
4, [1]
5, [1, 1]
6, [2]
7, [3]
8, [1]
9, [1, 1]
10, [1, 1]
11, [2, 1]
12, [2]
13, [1, 2]
14, [3]
15, [4]
16, [1]
17, [1, 1]
18, [1, 1]
19, [2, 1]
20, [1, 1]
21, [1, 1, 1]

Reinhard Zumkeller, Rows n = 0..1000 of triangle, flattened
Index entries for sequences related to binary expansion of n

Row sums = A000120 (the binary weight).
Cf. A245562, A071053.
Cf. A030308, A227736.
Row lengths are A069010.
The version for prime indices (instead of binary indices) is A124010.
Numbers with distinct run-lengths are A328592.
Numbers with equal run-lengths are A164707.
Cf. A003714, A014081, A328166, A328594, A328595, A328596.

import Data.List (group)
a245563 n k = a245563_tabf !! n !! k
a245563_row n = a245563_tabf !! n
a245563_tabf = [0] : map
   (map length . (filter ((== 1) . head)) . group) (tail a030308_tabf)
-- Reinhard Zumkeller, Aug 10 2014

for n from 0 to 128 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:=[op(lis),c]; out1:=1; c:=0;
fi;
if i = L1 and c>0 then lis:=[op(lis),c]; fi;
od:
lprint(n,lis);
od:

Join@@Table[Length/@Split[Join@@Position[Reverse[IntegerDigits[n,2]],1],#2==#1+1&],{n,0,100}] (* Gus Wiseman, Nov 03 2019 *)

from re import split
A245563_list = [0]
for n in range(1,100):
    A245563_list.extend(len(d) for d in split('0+',bin(n)[:1:-1]) if d != '')
# Chai Wah Wu, Sep 07 2014

A246029 a(n) = Product_{i in row n of A245562} prime(i).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 3, 5, 2, 4, 4, 6, 3, 6, 5, 7, 2, 4, 4, 6, 4, 8, 6, 10, 3, 6, 6, 9, 5, 10, 7, 11, 2, 4, 4, 6, 4, 8, 6, 10, 4, 8, 8, 12, 6, 12, 10, 14, 3, 6, 6, 9, 6, 12, 9, 15, 5, 10, 10, 15, 7, 14, 11, 13, 2, 4, 4, 6, 4, 8, 6, 10, 4, 8, 8, 12, 6, 12, 10, 14, 4, 8, 8, 12, 8, 16, 12, 20, 6, 12, 12, 18

Offset: 0

N. J. A. Sloane, Aug 15 2014; revised Sep 05 2014

nonn

This is the Run Length Transform of S(n) = {1,2,3,5,7,11,...} (1 followed by the primes).

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).

			From _Omar E. Pol_, Feb 12 2015: (Start)
Written as an irregular triangle in which row lengths is A011782:
1;
2;
2,3;
2,4,3,5;
2,4,4,6,3,6,5,7;
2,4,4,6,4,8,6,10,3,6,6,9,5,10,7,11;
2,4,4,6,4,8,6,10,4,8,8,12,6,12,10,14,3,6,6,9,6,12,9,15,5,10,10,15,7,14,11,13;
...
Right border gives the noncomposite numbers. This is simply a restatement of the theorem that this sequence is the Run Length Transform of A008578.
(End)

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 [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. A245562, A000045, A001045, A005940, A008578, A011782, A071053, A181819, A245565, A246028, A253081 (partial sums).

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(ithprime(i), i in lis);
ans:=[op(ans),a];
od:
ans;

f[n_, i_, x_] := f[n, i, x] = Which[n == 0, x, EvenQ[n], f[n/2, i + 1, x], True, f[(n - 1)/2, i, x*Prime[i]]];
a5940[n_] := f[n - 1, 1, 1];
a181819[n_] := Times @@ Prime[FactorInteger[n][[All, 2]]];
a[0] = 1; a[n_] := a181819[a5940[n + 1]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Aug 19 2018, after Antti Karttunen *)

from operator import mul
from functools import reduce
from re import split
from sympy import prime
def A246029(n):
    return reduce(mul,(prime(len(d)) for d in split('0+',bin(n)[2:]) if d != '')) if n > 0 else 1
# Chai Wah Wu, Sep 12 2014

a(n) = A181819(A005940(1+n)). - Antti Karttunen, Oct 15 2016

A246660 Run Length Transform of factorials.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 2, 2, 2, 6, 24, 1, 1, 1, 2, 1, 1, 2, 6, 2, 2, 2, 4, 6, 6, 24, 120, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 2, 2, 2, 6, 24, 2, 2, 2, 4, 2, 2, 4, 12, 6, 6, 6, 12, 24, 24, 120, 720, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 2, 2, 2, 6, 24, 1, 1, 1

Offset: 0

Peter Luschny, Sep 07 2014

For the definition of the Run Length Transform see A246595.

Only Jordan-Polya numbers (A001013) are terms of this sequence.

Antti Karttunen, Table of n, a(n) for n = 0..8192
Index entries for sequences computed with run length transform

Cf. A000142, A001013, A007814, A112624, A323505, A323506.
Cf. A003714 (gives the positions of ones).
Run Length Transforms of other sequences: A001316, A071053, A227349, A246588, A246595, A246596, A246661, A246674.

Table[Times @@ (Length[#]!&) /@ Select[Split[IntegerDigits[n, 2]], #[[1]] == 1&], {n, 0, 83}] (* Jean-François Alcover, Jul 11 2017 *)

A246660(n) = { my(i=0, p=1); while(n>0, if(n%2, i++; p = p * i, i = 0); n = n\2); p; };
for(n=0, 8192, write("b246660.txt", n, " ", A246660(n)));
\\ Antti Karttunen, Sep 08 2014

from operator import mul
from functools import reduce
from re import split
from math import factorial
def A246660(n):
    return reduce(mul,(factorial(len(d)) for d in split('0+',bin(n)[2:]) if d)) if n > 0 else 1 # Chai Wah Wu, Sep 09 2014

def RLT(f, size):
    L = lambda n: [a for a in Integer(n).binary().split('0') if a != '']
    return [mul([f(len(d)) for d in L(n)]) for n in range(size)]
A246660_list = lambda len: RLT(factorial, len)
A246660_list(88)

;; A stand-alone loop version, like the Pari-program above:
(define (A246660 n) (let loop ((n n) (i 0) (p 1)) (cond ((zero? n) p) ((odd? n) (loop (/ (- n 1) 2) (+ i 1) (* p (+ 1 i)))) (else (loop (/ n 2) 0 p)))))
;; One based on given recurrence, utilizing memoizing definec-macro from my IntSeq-library:
(definec (A246660 n) (cond ((zero? n) 1) ((even? n) (A246660 (/ n 2))) (else (* (A007814 (+ n 1)) (A246660 (/ (- n 1) 2))))))
;; Yet another implementation, using fold:
(define (A246660 n) (fold-left (lambda (a r) (* a (A000142 r))) 1 (bisect (reverse (binexp->runcount1list n)) (- 1 (modulo n 2)))))
(definec (A000142 n) (if (zero? n) 1 (* n (A000142 (- n 1)))))
;; Other functions are as in A227349 - Antti Karttunen, Sep 08 2014

a(2^n-1) = n!.
a(0) = 1, a(2n) = a(n), a(2n+1) = a(n) * A007814(2n+2). - Antti Karttunen, Sep 08 2014
a(n) = A112624(A005940(1+n)). - Antti Karttunen, May 29 2017
a(n) = A323505(n) / A323506(n). - Antti Karttunen, Jan 17 2019

A038184 State of one-dimensional cellular automaton 'sigma' (Rule 150): 000,001,010,011,100,101,110,111 -> 0,1,1,0,1,0,0,1 at generation n, converted to a decimal number.

Original entry on oeis.org

1, 7, 21, 107, 273, 1911, 5189, 28123, 65793, 460551, 1381653, 7039851, 17829905, 124809335, 340873541, 1840690907, 4295032833, 30065229831, 90195689493, 459568513131, 1172543963409, 8207807743863, 22286925370437

Offset: 0

Antti Karttunen, Feb 15 1999

nonn

Generation n (starting from the generation 0: 1) interpreted as a binary number, but written in base 10.
Rows of the mod 2 trinomial triangle (A027907), interpreted as binary numbers: 1, 111, 10101, 1101011, ... (A118110). - Jacob A. Siehler, Aug 25 2006
See A071053 for number of ON cells. - N. J. A. Sloane, Jul 28 2014

			Bit patterns with "0" replaced by "." for visibilty [_Georg Fischer_, Dec 16 2021]:
  0:                    1
  1:                   111
  2:                  1.1.1
  3:                 11.1.11
  4:                1...1...1
  5:               111.111.111
  6:              1.1...1...1.1
  7:             11.11.111.11.11
  8:            1.......1.......1
  9:           111.....111.....111
  10:         1.1.1...1.1.1...1.1.1
  11:        11.1.11.11.1.11.11.1.11
  12:       1...1.......1.......1...1
  13:      111.111.....111.....111.111
  14:     1.1...1.1...1.1.1...1.1...1.1
  15:    11.11.11.11.11.1.11.11.11.11.11

Gheorghe Coserea, Table of n, a(n) for n = 0..200
Alan J. Macfarlane, On generating functions of some sequences of integers defined in the evolution of the cellular automaton Rule 150, Preprint 2016.
Eric Weisstein's World of Mathematics, Rule 150
Index entries for sequences related to cellular automata

Cf. A006977, A006978, A038183, A038185 (other cellular automata).
Cf. A048710, A048720, A027907, A001317, A071053.
This sequence, A071036 and A118110 are equivalent descriptions of the Rule 150 automaton.

bit_n := (x,n) -> `mod`(floor(x/(2^n)),2);
sigmagen := proc(n) option remember: if (0 = n) then (1)
else sum('((bit_n(sigmagen(n-1),i)+bit_n(sigmagen(n-1),i-1)+bit_n(sigmagen(n-1),i-2)) mod 2)*(2^i)', 'i'=0..(2*n)) fi: end:

f[n_] := Sum[2^k*Coefficient[ #, x, k], {k, 0, 2n}] & @ Expand[(1 + x + x^2)^n, Modulus -> 2] (* Jacob A. Siehler, Aug 25 2006 *)

a(n) = subst(lift(Pol(Mod([1,1,1],2),'x)^n),'x,2);
vector(23,n,a(n-1))  \\ Gheorghe Coserea, Jun 12 2016

A246035 Number of odd terms in f^n, where f = (1/x+1+x)*(1/y+1+y).

Original entry on oeis.org

1, 9, 9, 25, 9, 81, 25, 121, 9, 81, 81, 225, 25, 225, 121, 441, 9, 81, 81, 225, 81, 729, 225, 1089, 25, 225, 225, 625, 121, 1089, 441, 1849, 9, 81, 81, 225, 81, 729, 225, 1089, 81, 729, 729, 2025, 225, 2025, 1089, 3969, 25, 225, 225, 625, 225, 2025, 625, 3025, 121, 1089, 1089, 3025, 441, 3969, 1849, 7225, 9, 81, 81, 225, 81, 729, 225

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.
This is the odd-rule cellular automaton defined by OddRule 777 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link).
Run Length Transform of {A001045(k+2)^2} (or of A139818).
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).

			Here is the neighborhood:
[X, X, X]
[X, X, X]
[X, X, X]
which contains a(1) = 9 ON cells.
.
From Omar E. Pol, Mar 17 2015: (Start)
Apart from the initial 1, the sequence can be written also as an irregular tetrahedron T(s,r,k) = A139818(r+2) * a(k), s>=1, 1<=r<=s, 0<=k<=(A011782(s-r)-1) as shown below:
..
9;
...
9;
25;
..........
9,     81;
25;
121;
....................
9,     81,  81, 225;
25,   225;
121;
441;
........................................
9,     81,  81, 225, 81, 729, 225, 1089;
25,   225, 225, 625;
121, 1089;
441;
1849;
...
Note that every row r is equal to A139818(r+2) times the beginning of the sequence itself, thus in 3D every column contains the same number: T(s,r,k) = T(s+1,r,k).
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..8192
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, A246034.

Cf. A001045, A139818.

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)*(1/y+1+y);
OddCA(f, 70);

b[0] = 1; b[n_] := b[n] = Expand[b[n - 1]*(x^2 + x + 1)];
a[n_] := Count[CoefficientList[b[n], x], _?OddQ]^2;
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Apr 30 2017 *)

a(n) = A071053(n)^2.

A246588 Run Length Transform of S(n) = wt(n) = 0,1,1,2,1,2,2,3,1,... (cf. A000120).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1

Offset: 0

N. J. A. Sloane, Sep 05 2014

nonn

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).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A000120.
Cf. A167489, A030308.
Run Length Transforms of other sequences: A071053, A227349, A246595, A246596, A246660, A246661, A246674.

import Data.List (group)
a246588 = product . map (a000120 . length) .
          filter ((== 1) . head) . group . a030308_row
-- Reinhard Zumkeller, Feb 13 2015, Sep 05 2014

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;
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(wt(i), i in lis);
ans:=[op(ans), a];
od:
ans;

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

from operator import mul
from functools import reduce
from re import split
def A246588(n):
    return reduce(mul,(bin(len(d)).count('1') for d in split('0+',bin(n)[2:]) if d)) if n > 0 else 1 # Chai Wah Wu, Sep 07 2014

# uses[RLT from A246660]
A246588_list = lambda len: RLT(lambda n: sum(Integer(n).digits(2)), len)
A246588_list(88) # Peter Luschny, Sep 07 2014

A246595 Run Length Transform of squares.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 4, 9, 1, 1, 1, 4, 4, 4, 9, 16, 1, 1, 1, 4, 1, 1, 4, 9, 4, 4, 4, 16, 9, 9, 16, 25, 1, 1, 1, 4, 1, 1, 4, 9, 1, 1, 1, 4, 4, 4, 9, 16, 4, 4, 4, 16, 4, 4, 16, 36, 9, 9, 9, 36, 16, 16, 25, 36, 1, 1, 1, 4, 1, 1, 4, 9, 1, 1, 1, 4, 4, 4, 9, 16, 1, 1, 1, 4, 1, 1

Offset: 0

N. J. A. Sloane, Sep 06 2014

nonn

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).

			From _Omar E. Pol_, Feb 10 2015: (Start)
Written as an irregular triangle in which row lengths is A011782:
1;
1;
1,4;
1,1,4,9;
1,1,1,4,4,4,9,16;
1,1,1,4,1,1,4,9,4,4,4,16,9,9,16,25;
1,1,1,4,1,1,4,9,1,1,1,4,4,4,9,16,4,4,4,16,4,4,16,36,9,9,9,36,16,16,25,36;
...
Right border gives A253909: 1 together with the positive squares.
(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;
..
1;
..
1;
4;
.......
1,   1;
4;
9;
...............
1,   1,  1,  4;
4,   4;
9;
16;
.............................
1,   1,  1,  4, 1, 1,  4,  9;
4,   4,  4, 16;
9,   9;
16;
25;
......................................................
1,   1,  1,  4, 1, 1,  4,  9, 1, 1, 1, 4, 4, 4, 9, 16;
4,   4,  4, 16, 4, 4, 16, 36;
9,   9,  9, 36;
16, 16;
25;
36;
...
Apart from the initial 1, we have that T(s,r,k) = T(s+1,r,k).
(End)

Chai Wah Wu, Table of n, a(n) for n = 0..8192
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.

Cf. A000290, A253082.
Cf. A003714 (gives the positions of ones).
Run Length Transforms of other sequences: A071053, A227349, A246588, A246596, A246660, A246661, A246674.

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(i^2, i in lis);
ans:=[op(ans), a];
od:
ans;

Table[Times @@ (Length[#]^2&) /@ Select[Split[IntegerDigits[n, 2]], #[[1]] == 1&], {n, 0, 85}] (* Jean-François Alcover, Jul 11 2017 *)

from operator import mul
from functools import reduce
from re import split
def A246595(n):
    return reduce(mul,(len(d)**2 for d in split('0+',bin(n)[2:]) if d != '')) if n > 0 else 1 # Chai Wah Wu, Sep 07 2014

# uses[RLT from A246660]
A246595_list = lambda len: RLT(lambda n: n^2, len)
A246595_list(86) # Peter Luschny, Sep 07 2014

; using MIT/GNU Scheme
(define (A246595 n) (fold-left (lambda (a r) (* a r r)) 1 (bisect (reverse (binexp->runcount1list n)) (- 1 (modulo n 2)))))
;; Other functions are as in A227349 - Antti Karttunen, Sep 08 2014

a(n) = A227349(n)^2. - Omar E. Pol, Feb 10 2015

A246596 Run Length Transform of Catalan numbers A000108.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 5, 1, 1, 1, 2, 2, 2, 5, 14, 1, 1, 1, 2, 1, 1, 2, 5, 2, 2, 2, 4, 5, 5, 14, 42, 1, 1, 1, 2, 1, 1, 2, 5, 1, 1, 1, 2, 2, 2, 5, 14, 2, 2, 2, 4, 2, 2, 4, 10, 5, 5, 5, 10, 14, 14, 42, 132, 1, 1, 1, 2, 1, 1, 2, 5, 1, 1, 1, 2, 2, 2, 5, 14, 1, 1, 1, 2, 1, 1, 2, 5

Offset: 0

N. J. A. Sloane, Sep 06 2014

nonn

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).

			From _Omar E. Pol_, Feb 15 2015: (Start)
Written as an irregular triangle in which row lengths are the terms of A011782:
1;
1;
1,2;
1,1,2,5;
1,1,1,2,2,2,5,14;
1,1,1,2,1,1,2,5,2,2,2,4,5,5,14,42;
1,1,1,2,1,1,2,5,1,1,1,2,2,2,5,14,2,2,2,4,2,2,4,10,5,5,5,10,14,14,42,132;
...
Right border gives the Catalan numbers. This is simply a restatement of the theorem that this sequence is the Run Length Transform of A000108.
(End)

Cf. A000108.
Cf. A003714 (gives the positions of ones).
Run Length Transforms of other sequences: A005940, A069739, A071053, A227349, A246588, A246595, A246660, A246661, A246674.

Cat:=n->binomial(2*n,n)/(n+1);
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(Cat(i), i in lis);
ans:=[op(ans), a];
od:
ans;

f = CatalanNumber; Table[Times @@ (f[Length[#]]&) /@ Select[ Split[ IntegerDigits[n, 2]], #[[1]] == 1&], {n, 0, 87}] (* Jean-François Alcover, Jul 11 2017 *)

from operator import mul
from functools import reduce
from gmpy2 import divexact
from re import split
def A246596(n):
    s, c = bin(n)[2:], [1, 1]
    for m in range(1, len(s)):
        c.append(divexact(c[-1]*(4*m+2),(m+2)))
    return reduce(mul,(c[len(d)] for d in split('0+',s))) if n > 0 else 1
# Chai Wah Wu, Sep 07 2014

# uses[RLT from A246660]
A246596_list = lambda len: RLT(lambda n: binomial(2*n, n)/(n+1), len)
A246596_list(88) # Peter Luschny, Sep 07 2014

; using MIT/GNU Scheme
(define (A246596 n) (fold-left (lambda (a r) (* a (A000108 r))) 1 (bisect (reverse (binexp->runcount1list n)) (- 1 (modulo n 2)))))
(define A000108 (EIGEN-CONVOLUTION 1 *))
;; Note: EIGEN-CONVOLUTION can be found from my IntSeq-library and other functions are as in A227349. - Antti Karttunen, Sep 08 2014

a(n) = A069739(A005940(1+n)). - Antti Karttunen, May 29 2017

cp's OEIS Frontend

A253102 a(n) = A071053(n)^3.

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A245562 Table read by rows: row n gives list of lengths of runs of 1's in binary expansion of n, starting with high-order bits.

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A245563 Table read by rows: row n gives list of lengths of runs of 1's in binary expansion of n, starting with low-order bits.

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A246029 a(n) = Product_{i in row n of A245562} prime(i).

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A246660 Run Length Transform of factorials.

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A038184 State of one-dimensional cellular automaton 'sigma' (Rule 150): 000,001,010,011,100,101,110,111 -> 0,1,1,0,1,0,0,1 at generation n, converted to a decimal number.

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A246035 Number of odd terms in f^n, where f = (1/x+1+x)*(1/y+1+y).

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