A048705 -id:A048705 - cp's OEIS frontend

A048706 XOR-conjugate rules of 1-D cellular automata rules given in A048705.

150, 90, 2523490710, 199931532107794273605284333428918544790, 226413559313153607979257138616992421290

Offset: 1

Antti Karttunen, Mar 09 1999

nonn

In hexadecimal this sequence looks like: 96,5A,96696996,96696996699696696996966996696996,
AA55AA55AA55AA5555AA55AA55AA55AA,
9966669966999966996666996699996666999966996666996699996699666699669999 6699666699669999669966669999666699669999669966669966999966, ...

Index entries for sequences related to cellular automata

# Other procedures as with A048705
rule90x150combination_xored := proc(n) local r,d,p,q,j,s,k,pattern;
p := extended_A020652[ n ]; # the Rule 150 component [ 0,1,op(A020652) ]
q := extended_A020653[ n ]; # the Rule 90 component [ 1,0,op(A020653) ]
r := p+q; # radius of CA.
d := (2*r)+1; # diameter of CA, including the cell itself.
s := 0; for k from 0 to (2^d)-1 do if(bit_i(k,r) <> bit_i(rule90(rule150(k,p),q),(2*r))) then s := s + 2^k; fi; od; RETURN(s); end;

A048725 a(n) = Xmult(n,5) or rule90(n,1).

Original entry on oeis.org

0, 5, 10, 15, 20, 17, 30, 27, 40, 45, 34, 39, 60, 57, 54, 51, 80, 85, 90, 95, 68, 65, 78, 75, 120, 125, 114, 119, 108, 105, 102, 99, 160, 165, 170, 175, 180, 177, 190, 187, 136, 141, 130, 135, 156, 153, 150, 147, 240, 245, 250, 255, 228, 225, 238, 235, 216, 221, 210

Offset: 0

Antti Karttunen, Apr 26 1999

The orbit of 1 under iteration of this function is the Sierpinski gasket A038183. It is called "rule 90" because the 8 bits of 90 = 01011010 in binary give bit k of the result as function of the value in {0,...,7} made out of bits k,k+1,k+2 of the input (i.e., floor(input / 2^k) mod 8). - M. F. Hasler, Oct 09 2017

			   n (in binary) | 4n [binary] | n XOR 4n [binary] | [decimal] = a(n)
          0      |        0    |           0       |        0
          1      |      100    |         101       |        5
         10      |     1000    |        1010       |       10
         11      |     1100    |        1111       |       15
        100      |    10000    |       10100       |       20
        101      |    10100    |       10001       |       17
   etc.

Alois P. Heinz, Table of n, a(n) for n = 0..16383

Cf. A048720, A048705, A048710, A048724, A048727, A048729.
Cf. A038183.
Cf. A353167 (terms sorted).

a:= n-> Bits[Xor](n*4, n):
seq(a(n), n=0..120);  # Alois P. Heinz, Aug 24 2019

Table[ BitXor[4n, n], {n, 0, 60}] (* Robert G. Wilson v, Jul 06 2006 *)

a(n)=bitxor(n,4*n) \\ Charles R Greathouse IV, Oct 03 2016

def A048725(n): return n^ n<<2 # Chai Wah Wu, Jun 29 2022

a(n) = n XOR n*2 XOR (n XOR n*2)*2 = A048724(A048724(n)). - Reinhard Zumkeller, Nov 12 2004

a(n) = n XOR (4n). - M. F. Hasler, Oct 09 2017

A048727 a(n) = Xmult(n,7) or rule150(n,1).

Original entry on oeis.org

0, 7, 14, 9, 28, 27, 18, 21, 56, 63, 54, 49, 36, 35, 42, 45, 112, 119, 126, 121, 108, 107, 98, 101, 72, 79, 70, 65, 84, 83, 90, 93, 224, 231, 238, 233, 252, 251, 242, 245, 216, 223, 214, 209, 196, 195, 202, 205, 144, 151, 158, 153, 140, 139, 130, 133, 168, 175

Offset: 0

Antti Karttunen, Apr 26 1999

Sequence gives binary encodings of polynomials in maximal ideal generated by x^2 + x + 1 in the polynomial ring GF(2)[X]. E.g. 1 * x^2+x+1 = x^2 +x+1 = 111 (binary encoding) = 7 (in decimal) x * x^2+x+1 = x^3+x^2+x = 1110 = 14 x+1 * x^2+x+1 = x^3+1 = 1001 = 9 x^2 * x^2+x+1 = x^4+x^3+x^2 = 11100 = 28 x^2+1 * x^2+x+1 = x^4+x^3+x+1 = 11011 = 27 etc.

David A. Corneth, Table of n, a(n) for n = 0..8191

Cf. A048720, A048705, A048710, A048725, A048730.

A048727[n_] := BitXor[n, 2*n, 4*n];
Array[A048727, 100, 0] (* Paolo Xausa, Aug 06 2025 *)

a(n)=bitxor(n,bitxor(2*n,4*n)) \\ Charles R Greathouse IV, Oct 03 2016

def A048727(n): return n^ n<<1 ^ n<<2 # Chai Wah Wu, Jun 29 2022

A048710 Family 1 "Rule 90 x Rule 150 Array" read by antidiagonals.

Original entry on oeis.org

1, 5, 7, 17, 27, 21, 85, 119, 65, 107, 257, 427, 325, 455, 273, 1285, 1799, 1105, 1755, 1365, 1911, 4369, 6939, 5397, 7607, 4097, 6827, 5189, 21845, 30583, 16705, 27499, 20485, 28679, 17745, 28123, 65537

Offset: 0

Antti Karttunen, Mar 18 1999

Infinitely many one-dimensional cellular automaton rules (given in sequence A048705) occur in this array, as combinations of CA-rules "90" (generates rows) and "150" (generates columns).
No pattern occurs twice in such arrays.
Each row/column can be generated from its predecessor row/column with SHIFTXORADJ transformation, given in A048711.

			   1  5  17   85  257 1105 ... [ beginning of array ]
   7 27 119  427 1799 ...
  21 65 325 1105 5397 ...

Rows = A038183, A048711, A048713, columns = A038184, A048712, A048713, diagonal = A048709. Cf. A048720.

trinv := n -> floor((1+sqrt(1+8*n))/2); # Gives integral inverses of the triangular numbers

a(n) = rule150(rule90(1, (((trinv(n)-1)*(((1/2)*trinv(n))+1))-n)), (n-((trinv(n)*(trinv(n)-1))/2))).

A048708 Numerators of ratios converging to the Thue-Morse constant, converted to hexadecimal.

Original entry on oeis.org

0, 1, 6, 69, 6996, 69969669, 6996966996696996, 69969669966969969669699669969669, 6996966996696996966969966996966996696996699696696996966996696996

Offset: 0

Antti Karttunen, Mar 09 1999

Starting from the term 6, the next term can be obtained from the previous one by appending its "horizontally" flipped variant (6 <-> 9, which equals its binary complement). Also, starting from the term 69, every second term stays same if read upside down from the other end (i.e., those numbers look same even if you rotate them by 180 degrees).

Cf. A048705, A048706, A048707.

map(convert, A048707, hex)

A048709 Main diagonal of Family 1 "Rule 90 x Rule 150" array.

Original entry on oeis.org

1, 27, 325, 7607, 69649, 1749419, 22103317, 476952263, 4311744769, 116417108763, 1392727114821, 32619053820599, 300171238899985, 7506480532757163, 94597931458646037, 2049660569696039367

Offset: 0

Antti Karttunen, Mar 09 1999

nonn

The successive generations of one-dimensional cellular automata rule (k=1,r=2) 1721342310 (hex 66999966) starting from the seed pattern 1.

Index entries for sequences related to cellular automata

A038183, A038184 give the first row and column of Family 1. Cf. A048705.

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A048706 XOR-conjugate rules of 1-D cellular automata rules given in A048705.

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A048725 a(n) = Xmult(n,5) or rule90(n,1).

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A048727 a(n) = Xmult(n,7) or rule150(n,1).

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A048710 Family 1 "Rule 90 x Rule 150 Array" read by antidiagonals.

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A048708 Numerators of ratios converging to the Thue-Morse constant, converted to hexadecimal.

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A048709 Main diagonal of Family 1 "Rule 90 x Rule 150" array.

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