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Search: a060547 -id:a060547
Displaying 1-6 of 6 results found.
page 1
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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
+10
8
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
(list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,4
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
LINKS
R. H. Hardin, Table of n, a(n) for n = 1..180
EXAMPLE
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
CROSSREFS
Column 1 is A060547(n-2)
KEYWORD
nonn,tabl
AUTHOR
R. H. Hardin Oct 13 2011
STATUS
approved
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.
+10
2
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,7
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
Harry J. Smith, Table of n, a(n) for n = 1..500
A. Barbé, Symmetric patterns in the cellular automaton that generates Pascal's triangle modulo 2, Discr. Appl. Math. 105(2000), 1-38.
Index entries for sequences related to cellular automata
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.
Empirical 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
PROG
(PARI) { for (n=1, 500, write("b060550.txt", n, " ", 2^(floor(n/3) + (n%3)%2 - 1) - 2^(floor((n + 3)/6) + (n%6==1) - 1)); ) } \\ Harry J. Smith, Jul 07 2009
CROSSREFS
Cf. A060547, A060548, A008611, A008615.
KEYWORD
easy,nonn
AUTHOR
André Barbé (Andre.Barbe(AT)esat.kuleuven.ac.be), Apr 03 2001
STATUS
approved
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.
+10
1
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
LINKS
Harry J. Smith, Table of n, a(n) for n=1..500
A. Barbé, Symmetric patterns in the cellular automaton that generates Pascal's triangle modulo 2, Discr. Appl. Math. 105(2000), 1-38.
Index entries for sequences related to cellular automata
Index entries for linear recurrences with constant coefficients, signature (2,2,-2,-4,-4,10,-4,-4,4,8,8,-16).
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
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 *)
PROG
(PARI) { for (n=1, 500, a=2^n-3*2^ceil(n/2)-2^(floor(n/3)+(n%3)%2)+3*2^(floor((n+3)/6)+(n%6==1)); write("b060551.txt", n, " ", a); ) } \\ Harry J. Smith, Jul 07 2009
CROSSREFS
Cf. A000079, A060546, A060547, A060548, A008619, A008611, A008615.
KEYWORD
easy,nonn
AUTHOR
André Barbé (Andre.Barbe(AT)esat.kuleuven.ac.be), Apr 03 2001
EXTENSIONS
More terms from Colin Barker, Aug 29 2013
STATUS
approved
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.
+10
1
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,5
LINKS
Harry J. Smith, Table of n, a(n) for n=1..500
A. Barbé, Symmetric patterns in the cellular automaton that generates Pascal's triangle modulo 2, Discr. Appl. Math. 105(2000), 1-38.
Index entries for sequences related to cellular automata
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)
PROG
(PARI) { for (n=1, 500, a=(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); write("b060552.txt", n, " ", a); ) } \\ Harry J. Smith, Jul 07 2009
CROSSREFS
Cf. A000079, A060547, A060548, A060546, A060552, A008611, A008615, A008619.
KEYWORD
easy,nonn
AUTHOR
André Barbé (Andre.Barbe(AT)esat.kuleuven.ac.be), Apr 03 2001
STATUS
approved
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.
+10
1
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Harry J. Smith, Table of n, a(n) for n=1..500
A. Barbé, Symmetric patterns in the cellular automaton that generates Pascal's triangle modulo 2, Discr. Appl. Math. 105(2000), 1-38.
Index entries for sequences related to cellular automata
Index entries for linear recurrences with constant coefficients, signature (2,2,-2,-4,-4,8).
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
PROG
(PARI) { for (n=1, 500, a=(2^(n-1) + 2^(floor(n/3) + (n%3)%2))/3 + 2^floor((n-1)/2); write("b060553.txt", n, " ", a); ) } \\ Harry J. Smith, Jul 07 2009
CROSSREFS
Cf. A000079, A060547, A060546, A008611, A008619.
KEYWORD
easy,nonn
AUTHOR
André Barbé (Andre.Barbe(AT)esat.kuleuven.ac.be), Apr 03 2001
EXTENSIONS
More terms from Colin Barker, Aug 29 2013
STATUS
approved
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.
+10
1
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,5
COMMENTS
The n-th row has A060547(n) terms.
Every positive term of A334556, say m, appears in row A070939(m).
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..1275 (rows 1 <= n <= 24, flattened)
A. Barbé, Symmetric patterns in the cellular automaton that generates Pascal's triangle modulo 2, Discr. Appl. Math. 105(2000), 1-38.
Michael De Vlieger, Diagram montage showing 486 XOR-triangles T(n,k)>0 for 3 <= n <= 20.
Michael De Vlieger, Large 50X25 Diagram montage showing 1250 XOR-triangles T(n,k)>0 for 3 <= n <= 24.
Rémy Sigrist, Triangles illustrating the initial terms
Rémy Sigrist, PARI program for A335061
Index entries for sequences related to binary expansion of n
Index entries for sequences related to cellular automata
Index entries for sequences related to XOR-triangles
EXAMPLE
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
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 *)
PROG
(PARI) See Links section.
CROSSREFS
Cf. A060547 (row length), A070939, A334556.
KEYWORD
nonn,base,tabf
AUTHOR
Rémy Sigrist, May 21 2020
STATUS
approved
page 1

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