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.

A267816 Decimal representation of the n-th iteration of the "Rule 221" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 3, 23, 111, 479, 1983, 8063, 32511, 130559, 523263, 2095103, 8384511, 33546239, 134201343, 536838143, 2147418111, 8589803519, 34359476223, 137438429183, 549754765311, 2199021158399, 8796088827903, 35184363700223, 140737471578111, 562949919866879
Offset: 0

Views

Author

Robert Price, Jan 20 2016

Keywords

References

  • S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Links

Crossrefs

Cf. A267814.
Similar entries: A085601 (2^(n + 1) * (2^n + 1) + 1), A092431 (2^(n - 1) * (2^n + 1) - 1), A092440 (2^(n + 1) * (2^n - 1) + 1), A129868 (2^(n - 1) * (2^n - 1) - 1), A134169 (2^(n - 1) * (2^n - 1) + 1), A281481 (2^(n - 1) * (2^n + 1) + 1), A281482 (2^(n + 1) * (2^n + 1) - 1). - Jaroslav Krizek, Jan 22 2017

Programs

  • Mathematica
    rule=221; rows=20; ca=CellularAutomaton[rule,{{1},0},rows-1,{All,All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]],{rows-k+1,rows+k-1}],{k,1,rows}]; (* Truncated list of each row *) Table[FromDigits[catri[[k]],2],{k,1,rows}]   (* Decimal Representation of Rows *)

Formula

Conjectures from Colin Barker, Jan 22 2016 and Apr 16 2019: (Start)
a(n) = 7*a(n-1)-14*a(n-2)+8*a(n-3) for n>3.
G.f.: (1-4*x+16*x^2-16*x^3) / ((1-x)*(1-2*x)*(1-4*x)).
(End)
a(n) = 2^(n + 1) * (2^n - 1) - 1, for n > 0. - Jaroslav Krizek, Jan 22 2017

A281481 a(n) = 2^(n - 1) * (2^n + 1) + 1.

Original entry on oeis.org

2, 4, 11, 37, 137, 529, 2081, 8257, 32897, 131329, 524801, 2098177, 8390657, 33558529, 134225921, 536887297, 2147516417, 8590000129, 34359869441, 137439215617, 549756338177, 2199024304129, 8796095119361, 35184376283137, 140737496743937, 562949970198529
Offset: 0

Views

Author

Jaroslav Krizek, Jan 22 2017

Keywords

Links

Crossrefs

Similar sequences: A085601 (2^(n + 1) * (2^n + 1) + 1), A092431 (2^(n - 1) * (2^n + 1) - 1), A092440 (2^(n + 1) * (2^n - 1) + 1), A129868 (2^(n - 1) * (2^n - 1) - 1), A134169 (2^(n - 1) * (2^n - 1) + 1), A267816 (2^(n + 1) * (2^n - 1) - 1), A281482 (2^(n + 1) * (2^n + 1) - 1).
Cf. A278930.

Programs

  • Magma
    [2^(n - 1) * (2^n + 1) + 1: n in [0..200]];
  • PARI
    Vec((2 - 10*x + 11*x^2) / ((1 - x)*(1 - 2*x)*(1 - 4*x)) + O(x^30)) \\ Colin Barker, Jan 22 2017

Formula

From Colin Barker, Jan 22 2017: (Start)
a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3) for n>2.
G.f.: (2 - 10*x + 11*x^2) / ((1 - x)*(1 - 2*x)*(1 - 4*x)).
(End)
a(n) = A278930(n - 2) for n >= 7. - Georg Fischer, Mar 26 2019

A335077 a(n) sets a record for side length k of zero-triangle in a rotationally symmetrical XOR-triangle.

Original entry on oeis.org

1, 11, 39, 543, 2391, 9559, 38231, 152919, 611671, 2446679, 9786711, 39146839, 156587351, 626349399, 2505397591, 10021590359, 40086361431, 160345445719, 641381782871, 2565527131479, 10262108525911, 41048434103639, 164193736414551, 656774945658199, 2627099782632791
Offset: 1

Views

Author

Michael De Vlieger, May 24 2020

Keywords

Comments

An XOR-triangle T(m) is an inverted 0-1 triangle formed by choosing a top row the binary rendition b(m) of m and having each entry in subsequent rows be the XOR of the two values above it, i.e., A038554(n) applied recursively until we reach a single bit. We may plot T(m) as an equilateral triangle, since each iteration decrements the binary integer length L of the output until we have L = 1.
The XOR function used here requires two inputs; if the inputs agree, the output is 0, else 1.
A rotationally-symmetrical XOR-triangle (RST) is one whose appearance is the same when rotated 120 degrees.
A zero triangle of side length k arises when we have a run of (k + 1) 1s in the preceding iteration.
This sequence contains m that produce T(m) with a recordsetting side length of its largest zero-triangle. For 1 < n < 3, T(a(n)) only has eccentric zero triangles. T(a(4)) has a singleton zero at center, thus a central zero triangle (CZT) of k = 1. For n > 4, all T(a(n)) have CZTs.
The number 543 = A281482(4); we observe that A281482(2^i) produces RSTs, and only for 0 <= i <= 2 do we have eccentric zero triangles larger than any possible CZT. For A281482(2^3) = 131583, the side length of its eccentric zero triangles prove much smaller than the largest possible CZT.
Since this sequence wants to maximize the side length k of the largest triangle, we see that the largest triangle possible is the CZT. Let j be the "frame width" or number of iterations required to generate the first run of 0s in the CZT. We note j >= 2, since j = 1 requires a run of (k + 1) ones delimited by at least 1 zero; such a width implies that these zeros occur at the beginning and end of b(m). However, beginning binary notation with a leading zero is not permitted. Therefore, if it is possible, we will see T(m) with j > 1.
The numbers that produce record-setting m are the smaller of the binary reverse of m, therefore binary weight favors the least significant digits. Thus we see a 1 followed by a number of zeros in a "prefix" A that, along with a suffix C, must have the same number of bits.
For RSTs with a CZT, we have only one way to produce a solid run of (k + 1) zeros, that is, by dithering bits, which necessitates paired 0 and 1, therefore, we have even k for n > 4.
A run-length formula for a(n) with n > 4 is 12..i(11)..3, meaning that we have 1 one, 2 zeros, any number i of paired 1-0 bits, and a run of 3 ones. Aside from the reversal of this pattern, which puts a greater binary weight in the most significant 3 bits, there is no other way to construct a smaller (or any) CZT with frame size j = 2.
This equates to linear recurrence kernel (5, -4) starting with {2391, 9559} (though 39 is part of the same trajectory).

Examples

			XOR-triangles T(a(n)) for 2 <= n <= 5, with "." = 0, "@" = 1:
                               a(4) = 543
                a(3) = 39   @ . . . . @ @ @ @ @
   a(2) = 11                 @ . . . @ . . . .
               @ . . @ @ @    @ . . @ @ . . .
    @ . @ @     @ . @ . .      @ . @ . @ . .
     @ @ .       @ @ @ .        @ @ @ @ @ .
      . @         . . @          . . . . @
       @           . @            . . . @
                    @              . . @
                                    . @
         a(5) = 2391                 @
    @ . . @ . @ . @ . @ @ @
     @ . @ @ @ @ @ @ @ . .
      @ @ . . . . . . @ .
       . @ . . . . . @ @
        @ @ . . . . @ .
         . @ . . . @ @
          @ @ . . @ .
           . @ . @ @
            @ @ @ .
             . . @
              . @
               @
.
          a(6) = 9559
  @ . . @ . @ . @ . @ . @ @ @
   @ . @ @ @ @ @ @ @ @ @ . .
    @ @ . . . . . . . . @ .
     . @ . . . . . . . @ @
      @ @ . . . . . . @ .
       . @ . . . . . @ @
        @ @ . . . . @ .
         . @ . . . @ @
          @ @ . . @ .
           . @ . @ @
            @ @ @ .
             . . @
              . @
               @

Links

Crossrefs

Cf. A038554, A070939, A281482, A334556, A334769, A334770, A334796, A334836.

Programs

  • Mathematica
    With[{s = Rest[Import["https://oeis.org/A334769/b334769.txt", "Data"][[All, -1]] ]}, Map[With[{w = NestWhileList[Map[BitXor @@ # &, Partition[#, 2, 1]] &, IntegerDigits[#, 2], Length@ # > 1 &]}, If[Length@ # == 0, 1, Times @@ Flatten@ MapIndexed[Prime[#2]^#1 &, #] &@ ReplacePart[ConstantArray[0, Max@ #[[All, 1]]], Map[#1 -> #2 & @@ # &, #]]] &@ Tally@ Flatten@ Array[If[# == 1, Map[If[First@ # == 1, Nothing, Length@ #] &, Split@ w[[#]] ], Map[If[First@ # == -1, Length@ #, Nothing] &, Split[w[[#]] - Most@ w[[# - 1]] ] ]] &, Length@ w]] /. -Infinity -> 0 &, s[[1 ;; 30]] ] ]
(* or *)
Join[{1}, Array[2^(# + 1)*(2^# + 1) - 1 &, 4, 1], LinearRecurrence[{5, -4}, {2391, 9559}, 21]]
(* or *)
Rest@ CoefficientList[Series[(12 - 27 x)/(12 x^2 - 15 x + 3) - 56 x^4 - 112 x^3 - 28 x^2 - 10 x - 4, {x, 0, 25}], x] (* Michael De Vlieger, Jun 01 2020 *)

Formula

a(1) = 1; for 2 <= n <= 4, a(n) = A281482(2^(n - 2)); a(5) = 2391, a(6) = 9559, for n > 6, a(n) = 5*a(n-1) - 4*a(n-2).
From Alejandro J. Becerra Jr., Jun 01 2020 : (Start)
For n > 4, a(n) = (5/3) + (7/3)*4^n.
G.f.: (4 - 9*x)/(4*x^2 - 5*x + 1) - 56*x^4 - 112*x^3 - 28*x^2 - 10*x - 4. (End)
Showing 1-3 of 3 results.

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