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 1-6 of 6 results.

A197317 T(n,k)=Number of nXk 0..4 arrays with each element x equal to the number of its horizontal and vertical neighbors equal to 0,1,1,0,0 for x=0,1,2,3,4.

Original entry on oeis.org

1, 1, 1, 2, 0, 2, 1, 3, 3, 1, 2, 2, 5, 2, 2, 4, 9, 10, 10, 9, 4, 2, 6, 128, 160, 128, 6, 2, 4, 27, 79, 152, 152, 79, 27, 4, 8, 18, 249, 790, 1033, 790, 249, 18, 8, 4, 83, 662, 2724, 4780, 4780, 2724, 662, 83, 4, 8, 56, 2767, 6242, 24903, 24704, 24903, 6242, 2767, 56, 8, 16, 257
Offset: 1

Views

Author

R. H. Hardin Oct 13 2011

Keywords

Comments

Every 0 is next to 0 0's, every 1 is next to 1 1's, every 2 is next to 2 1's, every 3 is next to 3 0's, every 4 is next to 4 0's
Table starts
.1..1....2.....1.......2........4..........2...........4.............8
.1..0....3.....2.......9........6.........27..........18............83
.2..3....5....10.....128.......79........249.........662..........2767
.1..2...10...160.....152......790.......2724........6242.........26422
.2..9..128...152....1033.....4780......24903......113774........553807
.4..6...79...790....4780....24704.....189212.....1400102.......8813744
.2.27..249..2724...24903...189212....2241018....20425821.....208960627
.4.18..662..6242..113774..1400102...20425821...282284587....3980881442
.8.83.2767.26422..553807..8813744..208960627..3980881442...82874179361
.4.56.3969.91756.2751427.62844698.1984221109.55455223337.1594105273961

Examples

			Some solutions containing all values 0 to 4 for n=6 k=4
..0..1..1..2....0..1..1..0....1..1..2..0....0..1..1..0....0..1..1..0
..3..0..2..1....1..2..0..3....0..2..1..1....3..0..2..1....3..0..2..1
..0..4..0..1....1..0..4..0....3..0..3..0....0..4..0..1....0..4..0..1
..3..0..3..0....0..4..0..1....0..4..0..1....3..0..3..0....1..0..4..0
..0..2..1..1....3..0..2..1....3..0..2..1....0..2..1..1....1..2..0..3
..1..1..2..0....0..1..1..0....0..1..1..2....1..1..2..0....2..1..1..0

Links

Crossrefs

Column 1 is A060547(n-2)

A060550 a(n) is the number of distinct patterns (modulo geometric D_3-operations) with no other than strict 120-degree rotational symmetry which can be formed by an equilateral triangular arrangement of closely packed black and white cells satisfying the local matching rule of Pascal's triangle modulo 2, where n is the number of cells in each edge of the arrangement.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 2, 1, 2, 6, 2, 6, 12, 6, 12, 28, 12, 28, 56, 28, 56, 120, 56, 120, 240, 120, 240, 496, 240, 496, 992, 496, 992, 2016, 992, 2016, 4032, 2016, 4032, 8128, 4032, 8128, 16256, 8128, 16256, 32640, 16256, 32640, 65280, 32640
Offset: 1

Views

Author

André Barbé (Andre.Barbe(AT)esat.kuleuven.ac.be), Apr 03 2001

Keywords

Comments

The matching rule is such that any elementary top-down triangle of three neighboring cells in the arrangement contains either one or three white cells.

Links

Crossrefs

Cf. A060547, A060548, A008611, A008615.

Programs

  • PARI
    a(n) = { 2^(floor(n/3) + (n%3)%2 - 1) - 2^(floor((n + 3)/6) + (n%6==1) - 1) } \\ Harry J. Smith, Jul 07 2009

Formula

a(n) = 2^(floor(n/3) + (n mod 3) mod 2 - 1) - 2^(floor((n+3)/6) + d(n)-1), with d(n)=1 if n mod 6=1, otherwise d(n)=0.
a(n) = (A060547(n) - A060548(n))/2.
a(n) = 2^(A008611(n-1) - 1) + 2^(A008615(n+1) - 1), for n >= 1.
G.f.: x^4*(x^2 - x + 1)*(x^2 + x + 1) / ((2*x^3-1)*(2*x^6-1)). - Colin Barker, Aug 29 2013

A060552 a(n) is the number of distinct (modulo geometric D3-operations) nonsymmetric (no reflective nor rotational symmetry) patterns which can be formed by an equilateral triangular arrangement of closely packed black and white cells satisfying the local matching rule of Pascal's triangle modulo 2, where n is the number of cells in each edge of the arrangement. The matching rule is such that any elementary top-down triangle of three neighboring cells in the arrangement contains either one or three white cells.

Original entry on oeis.org

0, 0, 0, 1, 2, 7, 14, 35, 70, 154, 310, 650, 1300, 2666, 5332, 10788, 21588, 43428, 86856, 174244, 348488, 697992, 1396040, 2794120, 5588240, 11180680, 22361360, 44730896, 89462032, 178940432, 357880864, 715794960
Offset: 1

Views

Author

André Barbé (Andre.Barbe(AT)esat.kuleuven.ac.be), Apr 03 2001

Keywords

Links

Crossrefs

Cf. A000079, A060547, A060548, A060546, A060552, A008611, A008615, A008619.

Programs

  • PARI
    a(n) = { (2^(n-1)-2^(floor(n/3)+(n%3)%2-1))/3+2^(floor((n+3)/6)+(n%6==1)-1)-2^floor((n-1)/2) } \\ Harry J. Smith, Jul 07 2009

Formula

a(n) = (2^(n-1) - 2^(floor(n/3) + (n mod 3)mod 2 - 1))/3 + 2^(floor((n+3)/6) + d(n) - 1) - 2^floor((n-1)/2), with d(n)=1 if n mod 6=1 else d(n)=0.
a(n) = (A000079(n-1) - A060547(n)/2)/3 + A060548(n)/2 -A060546(n)/2.
a(n) = (A000079(n-1) - 2^(A008611(n-1) - 1))/3 + 2^(A008615(n+1) - 1) - 2^(A008619(n-1) - 1), n >= 1.
From R. J. Mathar, Aug 03 2009: (Start)
a(n) = 2*a(n-1) + 2*a(n-2) - 2*a(n-3) - 4*a(n-4) - 4*a(n-5) + 10*a(n-6) - 4*a(n-7) - 4*a(n-8) + 4*a(n-9) + 8*a(n-10) + 8*a(n-11) - 16*a(n-12).
G.f.: -x^4*(-1 - x^2 - x^4 + 2*x^3 + 2*x^5 + 2*x^6)/((2*x-1)*(2*x^2-1)*(2*x^3-1)*(2*x^6-1)). (End)

A060551 a(n) is the number of nonsymmetric patterns (no reflective, nor rotational symmetry) which may be formed by an equilateral triangular arrangement of closely packed black and white cells satisfying the local matching rule of Pascal's triangle modulo 2, where n is the number of cells in each edge of the arrangement. The matching rule is such that any elementary top-down triangle of three neighboring cells in the arrangement contains either one or three white cells.

Original entry on oeis.org

0, 0, 0, 6, 12, 42, 84, 210, 420, 924, 1860, 3900, 7800, 15996, 31992, 64728, 129528, 260568, 521136, 1045464, 2090928, 4187952, 8376240, 16764720, 33529440, 67084080, 134168160, 268385376, 536772192, 1073642592, 2147285184, 4294769760, 8589539520, 17179472064
Offset: 1

Views

Author

André Barbé (Andre.Barbe(AT)esat.kuleuven.ac.be), Apr 03 2001

Keywords

Links

Crossrefs

Cf. A000079, A060546, A060547, A060548, A008619, A008611, A008615.

Programs

  • Mathematica
    LinearRecurrence[{2,2,-2,-4,-4,10,-4,-4,4,8,8,-16},{0,0,0,6,12,42,84,210,420,924,1860,3900},40] (* Harvey P. Dale, Feb 01 2015 *)
  • PARI
    a(n) = { 2^n-3*2^ceil(n/2)-2^(floor(n/3)+(n%3)%2)+3*2^(floor((n+3)/6)+(n%6==1)) } \\ Harry J. Smith, Jul 07 2009

Formula

a(n) = 2^n - 3*2^ceiling(n/2) - 2^(floor(n/3)+(n mod 3)mod 2) + 3*2^(floor((n+3)/6) + d(n)), with d(n)=1 if n mod 6=1 else d(n)=0.
a(n) = A000079(n) - 3*A060546(n) - A060547(n) + 3*A060548(n).
a(n) = A000079(n) - 3*2^A008619(n-1) - 2^A008611(n-1) + 3*2^A008615(n+1), for n >= 1.
G.f.: -6*x^4*(2*x^6 + 2*x^5 - x^4 + 2*x^3 - x^2 - 1) / ((2*x-1)*(2*x^2-1)*(2*x^3-1)*(2*x^6-1)). - Colin Barker, Aug 29 2013
a(n) = 6*A060552(n). - Andrew Howroyd, Dec 24 2024

Extensions

More terms from Colin Barker, Aug 29 2013

A060553 a(n) is the number of distinct (modulo geometric D3-operations) patterns which can be formed by an equilateral triangular arrangement of closely packed black and white cells satisfying the local matching rule of Pascal's triangle modulo 2, where n is the number of cells in each edge of the arrangement. The matching rule is such that any elementary top-down triangle of three neighboring cells in the arrangement contains either one or three white cells.

Original entry on oeis.org

2, 2, 4, 6, 10, 16, 32, 52, 104, 192, 376, 720, 1440, 2800, 5600, 11072, 22112, 43968, 87936, 175296, 350592, 700160, 1400192, 2798336, 5596672, 11188992, 22377984, 44747776, 89495040, 178973696, 357947392, 715860992, 1431721984, 2863378432, 5726754816
Offset: 1

Views

Author

André Barbé (Andre.Barbe(AT)esat.kuleuven.ac.be), Apr 03 2001

Keywords

Links

Crossrefs

Cf. A000079, A060547, A060546, A008611, A008619.

Programs

  • PARI
    a(n) = { (2^(n-1) + 2^(floor(n/3) + (n%3)%2))/3 + 2^floor((n-1)/2) } \\ Harry J. Smith, Jul 07 2009

Formula

a(n) = (2^(n-1)+2^(floor(n/3) + (n mod 3)mod 2))/3 + 2^floor((n-1)/2).
a(n) = (A000079(n-1) + A060547(n))/3 + A060546(n)/2.
a(n) = (A000079(n-1) + 2^A008611(n-1))/3 + 2^(A008619(n-1) - 1), for n >= 1.
G.f.: -2*x*(4*x^5 + x^4 - x^3 - 2*x^2 - x + 1) / ((2*x-1)*(2*x^2-1)*(2*x^3-1)). - Colin Barker, Aug 29 2013

Extensions

More terms from Colin Barker, Aug 29 2013

A335061 Irregular table read by rows; n-th row corresponds to numbers in the range 0..2^n-1 whose binary expansion (possibly left-padded with 0's up to n binary digits) generates rotationally symmetric XOR-triangles.

Original entry on oeis.org

0, 1, 0, 0, 2, 0, 6, 11, 13, 0, 14, 0, 30, 39, 57, 0, 8, 54, 62, 83, 91, 101, 109, 0, 126, 151, 233, 0, 40, 92, 116, 138, 162, 214, 254, 0, 72, 140, 196, 314, 370, 438, 510, 543, 599, 659, 731, 805, 877, 937, 993, 0, 168, 854, 1022, 1379, 1483, 1589, 1693
Offset: 1

Views

Author

Rémy Sigrist, May 21 2020

Keywords

Comments

The n-th row has A060547(n) terms.
Every positive term of A334556, say m, appears in row A070939(m).

Examples

			The first rows are:
    0, 1
    0
    0, 2
    0, 6, 11, 13
    0, 14
    0, 30, 39, 57
    0, 8, 54, 62, 83, 91, 101, 109
The XOR-triangles corresponding to the 8 terms of row 7 are (with dots instead of 0's for clarity):
   T(7,1) = 0:        T(7,2) = 8:        T(7,3) = 54:       T(7,4) = 62,
   . . . . . . .      . . . 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
         .                  .                  .                  .
   T(7,5) = 83:       T(7,6) = 91:       T(7,7) = 101:      T(7,8) = 109:
   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 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. A060547 (row length), A070939, A334556.

Programs

  • Mathematica
    Table[Select[Range[0, 2^n - 1], Block[{k = #, w}, (Reverse /@ Transpose[#] /. -1 -> Nothing) == w &@ MapIndexed[PadRight[#, n, -1] &, Set[w, NestList[Map[BitXor @@ # &, Partition[#, 2, 1]] &, PadLeft[IntegerDigits[k, 2], n], n - 1]]]] &], {n, 12}] // Flatten (* Michael De Vlieger, May 24 2020 *)
  • PARI
    See Links section.
Showing 1-6 of 6 results.

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