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 1-3 of 3 results.

A006977 Cellular automaton with Rule 230: 000, 001, 010, 011, ..., 111 -> 0,1,1,0,0,1,1,1.

Original entry on oeis.org

1, 3, 5, 15, 23, 59, 93, 239, 375, 955, 1501, 3823, 6007, 15291, 24029, 61167, 96119, 244667, 384477, 978671, 1537911, 3914683, 6151645, 15658735, 24606583, 62634939, 98426333, 250539759, 393705335, 1002159035, 1574821341, 4008636143, 6299285367
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

More precisely, this is the orbit of the initial value 1 under this Rule 230, cf. A292682. The substitution 100 -> 0 ensures that the initial 1 never extends to the right. - M. F. Hasler, Oct 09 2017

Examples

			   n  |    a(n) [binary] | a(n) [decimal]
   0  |      ...01(0...) |   1
   1  |     ...011(0...) |   3   (From ...010.., using 001 -> 1, 010 -> 1, 100 -> 0)
   2  |    ...0101(0...) |   5   (001 -> 1, 011 -> 0, 110 -> 1, 100 -> 0)
   3  |   ...01111(0...) |  15   (001 -> 1, 010 -> 1, 101 -> 1, 110 -> 1, 100 -> 0)
   4  |  ...010111(0...) |  23   (001 -> 1, 011 -> 0, 111 -> 1, 110 -> 1)
   5  | ...0111011(0...) |  59   (patterns of both of the above combined)
   6  |...01011101(0...) |  93   (as above)

References

  • Marc LeBrun, personal communication.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A292682; A292680, A292681, A266178, A266179, A266180; ...

Programs

  • Mathematica
    FromDigits[#, 2] & /@ CellularAutomaton[230, {{1}, 0}, 32] (* Michael De Vlieger, Oct 09 2017 *)
  • PARI
    vector(90,i,a=if(i>1,A292682(a),1))

Formula

Numbers suggest a rational g.f. with denominator (1-x^4)(1-4x^2). - Ralf Stephan, Jun 09 2005
G.f. seems to be (1 + 3*x + x^2 + 3*x^3 + 2*x^4 - 4*x^5)/((1 - x^4)*(1 - 4*x^2)). - Vincenzo Librandi, Sep 11 2017

Extensions

Edited by M. F. Hasler, Oct 09 2017

A292681 Rule 6: (000, ..., 111) -> (0, 1, 1, 0, 0, 0, 0, 0), without extending to the right of input bit 0.

Original entry on oeis.org

0, 3, 6, 4, 12, 13, 8, 8, 24, 27, 26, 24, 16, 17, 16, 16, 48, 51, 54, 52, 52, 53, 48, 48, 32, 35, 34, 32, 32, 33, 32, 32, 96, 99, 102, 100, 108, 109, 104, 104, 104, 107, 106, 104, 96, 97, 96, 96, 64, 67, 70, 68, 68, 69, 64, 64, 64, 67, 66, 64, 64, 65, 64, 64, 192, 195, 198, 196, 204, 205, 200, 200, 216
Offset: 0

Views

Author

M. F. Hasler, Oct 09 2017

Keywords

Comments

This is A292680 / 2; the orbit of 1 under A292680 is A266180.
Rule 6 is the smallest rule which is even (otherwise infinitely many bits would be switched on at step 1, for any finite starting value) and nontrivial (i.e., does not lead to extinction nor simple reproduction, possibly shifted left or right, of a single-bit initial state).
In the context of elementary cellular automata, one would usually consider one further bit of output (i.e., the cell which has bit 0 of the input as left neighbor). This leads to A292680 which has all terms even, so it is natural to consider this variant.

Examples

			     n        |         a(n)
   0 =   0[2] |      0[2] =  0
   1 =   1[2] |     11[2] =  3 (bits below 001 and 01(0) are on)
   2 =  10[2] |    110[2] =  6 (1 below 001 and 010, 0 below 10(0))
   3 =  11[2] |    100[2] =  4 (1 below 001, 0 below 011 and 11(0))
   4 = 100[2] |   1100[2] = 12 (as n = 1 and n = 2, shifted right once more)
   5 = 101[2] |   1101[2] = 13 (1 below 001 and 010 (twice), 0 below 101)
   6 = 110[2] |   1000[2] =  8 (as n = 3, shifted right once)
   7 = 111[2] |   1000[2] =  8 (1 below 001, 0 below 011, 111 and 11(0)).

Links

Crossrefs

Cf. A292680, A266178, A266179, A266180, A019590.

Programs

  • PARI
    apply( A292681(n,r=6)=sum(i=0,logint(!n+n<<=1,2)+1,bittest(r,(n>>i)%8)<

A292682 Rule 230: (000, ..., 111) -> (0, 1, 1, 0, 0, 1, 1, 1), without extending to the right of input bit 0.

Original entry on oeis.org

0, 3, 6, 5, 12, 15, 10, 11, 24, 27, 30, 29, 20, 23, 22, 23, 48, 51, 54, 53, 60, 63, 58, 59, 40, 43, 46, 45, 44, 47, 46, 47, 96, 99, 102, 101, 108, 111, 106, 107, 120, 123, 126, 125, 116, 119, 118, 119, 80, 83, 86, 85, 92, 95, 90, 91, 88, 91, 94, 93, 92, 95, 94, 95, 192, 195, 198, 197, 204, 207
Offset: 0

Views

Author

M. F. Hasler, Oct 09 2017

Keywords

Comments

The orbit of 1 under this rule is A006977.
The substitution rules 000 -> 0 and 100 -> 0 ensure that no (even or odd) input value can ever extend / "propagate" to the right, therefore it is not required to consider the additional digit to the right of input bit 0 (i.e., the cell which would have this bit 0 as left neighbor), as one would usually do in the context of elementary cellular automata (cf., e.g., A292680 vs. A292681).

Examples

			     n        |         a(n)
   0 =   0[2] |      0[2] =  0
   1 =   1[2] |     11[2] =  3  (bits below 001 and 01(0) are on)
   2 =  10[2] |    110[2] =  6  (1 below 001 and 010, 0 below 10(0))
   3 =  11[2] |    101[2] =  5  (1 below 001 and 11(0), 0 below 011.)
   4 = 100[2] |   1100[2] = 12  (as n = 1 and n = 2, shifted left once more)
   5 = 101[2] |   1111[2] = 15  (1 below 001, 010 (twice) and 101)
   6 = 110[2] |   1010[2] = 10  (as n = 3, shifted left once)
   7 = 111[2] |   1011[2] = 11  (1 below 001, 111 and 11(0), 0 below 011).

Links

Crossrefs

Cf. A006977, A292680, A292681, A266178, A266179, A266180, A019590.

Programs

  • PARI
    apply( A292682(n,r=230)=sum(i=0,logint(!n+n<<=1,2)+1,bittest(r,bitand(n>>i,7))<
Showing 1-3 of 3 results.

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