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.

A333625 Read terms e = T(n,k) in A333624 as Product(prime(k)^e) for n in A334556.

Original entry on oeis.org

1, 8, 8, 27, 27, 216, 512, 216, 512, 648, 648, 686, 12096, 46656, 262144, 46656, 262144, 12096, 686, 192000, 139968, 192000, 139968, 1866240, 179712, 74088, 91125, 74088, 91125, 179712, 1866240, 343000, 1000000, 5832000, 4251528, 5832000, 80621568, 13824000, 1073741824
Offset: 1

Views

Author

Michael De Vlieger, May 13 2020

Keywords

Comments

Row a(n) of A067255 = row A334556(n) of A333624.
An XOR-triangle t(n) is an inverted 0-1 triangle formed by choosing a top row the binary rendition of n 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.
Let T(n,k) address the terms in the k-th position of row n in A333624.
This sequence encodes T(n,k) via A067255 to succinctly express the number of zero-triangles in A334556(n). To decode a(n) => A333624(A334556(n)), we use A067255(a(n)).

Examples

			Relationship of this sequence to A334556 and A333624:
       n A334556(n) a(n)  Row n of A333624
       -----------------------------------
       1     1        1   0
       2    11        8   3
       3    13        8   3
       4    39       27   0, 3
       5    57       27   0, 3
       6    83      216   3, 3
       7    91      512   9
       8   101      216   3, 3
       9   109      512   9
      10   151      648   3, 4
      11   233      648   3, 4
      12   543      686   1, 0, 0, 3
      13   599    12096   6, 3, 0, 1
      14   659    46656   6, 6
      15   731   262144   18
      16   805    46656   6, 6
      ...
Let b(n) = n written in binary and let L(n) = ceiling(log_2(n)) = A070939(n). Let => be a single iteration of XOR across pairs of bits in b(n). Let t(n) be the XOR triangle initiated by b(n).
a(1) = 0, since b(1) = 1 and row 1 of A333624 is {0}. Since the XOR triangle t(1) that results from a single 1-bit merely consists of that bit and since there are no zeros in the triangle t(1), we write the single term zero in row n of A333624. thus a(n) = prime(1)^0 = 2^0 = 1.
a(2) = 8 because row A334556(2) of A333624 (i.e., the 11th row) has {3}. b(11) = 1011 => 110 => 01 => 1 (a rotationally symmetrical t(11)). We have 3 isolated zeros thus row 11 of A333624 = {3}, therefore a(2) = prime(1)^3 = 2^3 = 8.
a(4) = 27 because row A334556(4) of A333624 (i.e., the 39th row) has {0, 3}. b(39) = 100111 => 10100 => 1110 => 001 => 01 => 1 (a rotationally symmetrical t(39)). We have 3 isolated triangles of zeros with edge length 2, thus row 39 of A333624 = {0, 3}, therefore a(4) = prime(1)^0 * prime(2)^3 = 2^0 * 3^3 = 27.

Links

Crossrefs

Cf. A038554, A067255, A070939, A333624, A334591, A334556.

Programs

  • Mathematica
    With[{s = Rest[Import["https://oeis.org/A334556/b334556.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 ;; 36]]]]

A334896 Read terms e = T(n,k) in A333624 as Product(prime(k)^e) for n in A334769.

Original entry on oeis.org

648, 648, 686, 12096, 12096, 686, 192000, 139968, 192000, 139968, 1866240, 179712, 179712, 1866240, 814968, 2101248, 102036672, 331776000, 102036672, 331776000, 2101248, 814968, 179712000, 4423680000, 1866240000, 131010048, 179712000, 4423680000, 1866240000, 131010048
Offset: 1

Views

Author

Michael De Vlieger, May 23 2020

Keywords

Comments

Row a(n) of A067255 = row A334769(n) of A333624.
An XOR-triangle t(n) is an inverted 0-1 triangle formed by choosing a top row the binary rendition of n 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.
Let T(n,k) address the terms in the k-th position of row n in A333624.
This sequence encodes T(n,k) via A067255 to succinctly express the number of zero-triangles in A334769(n). To decode a(n) => A333624(A334769(n)), we use A067255(a(n)).

Examples

			a(1) = 648, since b(A334769(1)) = b(151) = 10010111, which generates T(151) as shown below, replacing 1 with "@" and 0 with ".":
  @ . . @ . @ @ @
   @ . @ @ @ . .
    @ @ . . @ .
     . @ . @ @
      @ @ @ .
       . . @
        . @
         @
In this figure we see 3 zero-triangles of side length k = 1, and 4 of side length k = 2, therefore, T(1,1) = 3 and T(1,2) = 4. This becomes 2^3 * 3^4 = 8 * 81 = 648.
Relationship of this sequence to A334556 and A333624:
        n  A334769(n)  a(n)  Row n of A333624
      --------------------------------------
       1    151       648    3, 4
       2    233       648    3, 4
       3    543       686    1, 0, 0, 3
       4    599     12096    6, 3, 0, 1
       5    937     12096    6, 3, 0, 1
       6    993       686    1, 0, 0, 3
       7   1379    192000    9, 1, 3
       8   1483    139968    6, 7
       9   1589    192000    9, 1, 3
      10   1693    139968    6, 7
      11   2359   1866240    9, 6, 1
      12   2391    179712    9, 3, 0, 0, 0, 1
      13   3753    179712    9, 3, 0, 0, 0, 1
      14   3785   1866240    9, 6, 1
      15   8607    814968    3, 3, 0, 3, 1
      16   9559   2101248   12, 3, 0, 0, 0, 0, 0, 1
      ...

Links

Crossrefs

Cf. A038554, A067255, A070939, A333624, A334591, A334556, A334769.

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 ;; 29]]]]

A334771 a(n) = smallest m that generates a rotationally symmetrical XOR-triangle T(m) with a central triangle of zeros with side length n.

Original entry on oeis.org

543, 151, 2359, 599, 8607, 2391, 37687, 9559, 137631, 38231, 602935, 152919, 2202015, 611671, 9646903, 2446679, 35232159, 9786711, 154350391, 39146839, 563714463, 156587351, 2469606199, 626349399, 9019431327, 2505397591, 39513699127, 10021590359
Offset: 1

Views

Author

Michael De Vlieger, May 10 2020

Keywords

Comments

An XOR-triangle T(m) is an inverted 0-1 triangle formed by choosing a top row the binary rendition of n and having each entry in subsequent rows be the XOR of the two values above it, i.e., A038554(m) applied recursively until we reach a single bit.
A334556 is the sequence of rotationally symmetrical T(m).
A central zero-triangle (CZT) is a field of contiguous 0-bits with side length n in T(m) surrounded on all sides by a layer of 1 bits, and generally k > 1 bits of any parity. Alternatively, these might be referred to as "central bubbles".

Examples

			First 4 terms shown below, replacing 0 with “.” for clarity:
    a(1) = 543; T(543):
  1 . . . . 1 1 1 1 1
   1 . . . 1 . . . .
    1 . . 1 1 . . .
     1 . 1 . 1 . .
      1 1 1 1 1 .
       . . . . 1
        . . . 1
         . . 1
          . 1
           1
a(2) = 151; T(151):
  1 . . 1 . 1 1 1
   1 . 1 1 1 . .
    1 1 . . 1 .
     . 1 . 1 1
      1 1 1 .
       . . 1
        . 1
         1
a(3) = 2359; T(2359):
  1 . . 1 . . 1 1 . 1 1 1
   1 . 1 1 . 1 . 1 1 . .
    1 1 . 1 1 1 1 . 1 .
     . 1 1 . . . 1 1 1
      1 . 1 . . 1 . .
       1 1 1 . 1 1 .
        . . 1 1 . 1
         . 1 . 1 1
          1 1 1 .
           . . 1
            . 1
             1
a(4) = 599; T(599):
  1 . . 1 . 1 . 1 1 1
   1 . 1 1 1 1 1 . .
    1 1 . . . . 1 .
     . 1 . . . 1 1
      1 1 . . 1 .
       . 1 . 1 1
        1 1 1 .
         . . 1
          . 1
           1

Links

Crossrefs

Cf. A038554, A070939, A334556, A334769, A334770.

Programs

  • Mathematica
    Block[{f, s = Rest[Import["https://oeis.org/A334556/b334556.txt", "Data"][[All, -1]] ], t, u}, f[n_] := NestWhileList[Map[BitXor @@ # &, Partition[#, 2, 1]] &, IntegerDigits[n, 2], Length@ # > 1 &]; Set[{t, u}, Transpose@ Array[Block[{n = s[[#]]}, If[# == 0, Nothing, {n, #}] &@ FirstCase[MapIndexed[If[2 #2 > #3 + 1, Nothing, #1[[#2 ;; -#2]]] & @@ {#1, First[#2], Length@ #1} &, f[n][[1 ;; Ceiling[IntegerLength[#, 2]/(2 Sqrt[3])] + 3]]  ], r_List /; FreeQ[r, 1] :> Length@ r] /. k_ /; MissingQ@ k -> 0] &, Length@ s - 1, 2]]; Array[If[! IntegerQ@ #, 0, t[[#]] ] &@ FirstPosition[u, #][[1]] &, Max@ u] ]
(* Second, more efficient program: *)
LinearRecurrence[{0, 0, 0, 17, 0, 0, 0, -16}, {543, 151, 2359, 599, 8607, 2391, 37687, 9559}, 28] (* Michael De Vlieger, May 20 2020 *)
  • PARI
    Vec(x*(543 + 151*x + 2359*x^2 + 599*x^3 - 624*x^4 - 176*x^5 - 2416*x^6 - 624*x^7) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)*(1 + x^2)*(1 + 4*x^2)) + O(x^30)) \\ Colin Barker, May 21 2020

Formula

a(n) = 17*a(n-4) - 16*a(n-8), starting with a(1) = 543, a(2) = 151, a(3) = 2359, a(4) = 599, a(5) = 8607, a(6) = 2391, a(7) = 37687, and a(8) = 9559.
G.f.: x*(543 + 151*x + 2359*x^2 + 599*x^3 - 624*x^4 - 176*x^5 - 2416*x^6 - 624*x^7) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)*(1 + x^2)*(1 + 4*x^2)). - Colin Barker, May 21 2020
Showing 1-3 of 3 results.

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