A056453 -id:A056453 - cp's OEIS frontend

A056463 Number of primitive (aperiodic) palindromes using exactly two different symbols.

0, 0, 2, 2, 6, 4, 14, 12, 28, 24, 62, 54, 126, 112, 246, 240, 510, 476, 1022, 990, 2030, 1984, 4094, 4020, 8184, 8064, 16352, 16254, 32766, 32484, 65534, 65280, 131006, 130560, 262122, 261576, 524286, 523264, 1048446, 1047540, 2097150, 2094988, 4194302, 4192254

Offset: 1

Marks R. Nester

nonn

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Column 2 of A327873.

Cf. A056453, A056458.

seq(n)={Vec(sum(k=1, n\3, moebius(k)*2*x^(3*k)/((1 - 2*x^(2*k))*(1 - x^k)) + O(x*x^n)), -n)} \\ Andrew Howroyd, Sep 29 2019

from sympy import mobius, divisors
def A056463(n): return sum(mobius(n//d)*((1<<(d+1>>1))-2) for d in divisors(n, generator=True)) # Chai Wah Wu, Feb 18 2024

a(n) = Sum_{d|n} mu(d)*A056453(n/d).

G.f.: Sum_{k>=1} mu(k)*2*x^(3*k)/((1 - 2*x^(2*k))*(1 - x^k)). - Andrew Howroyd, Sep 29 2019

Terms a(32) and beyond from Andrew Howroyd, Sep 28 2019

A285612 Binary representation of the diagonal from the corner to the origin of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 62", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 1, 10, 10, 110, 110, 1110, 1110, 11110, 11110, 111110, 111110, 1111110, 1111110, 11111110, 11111110, 111111110, 111111110, 1111111110, 1111111110, 11111111110, 11111111110, 111111111110, 111111111110, 1111111111110, 1111111111110, 11111111111110

Offset: 0

Robert Price, Apr 22 2017

Initialized with a single black (ON) cell at stage zero.

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Robert Price, Table of n, a(n) for n = 0..126
Robert Price, Diagrams of first 20 stages
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Wolfram Research, Wolfram Atlas of Simple Programs
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata

Cf. A285613, A056453, A233411.

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 62; stages = 128;
rule = IntegerDigits[code, 2, 10];
g = 2 * stages + 1; (* Maximum size of grid *)
a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
ca = a;
ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
PrependTo[ca, a];
(* Trim full grid to reflect growth by one cell at each stage *)
k = (Length[ca[[1]]] + 1)/2;
ca = Table[Table[Part[ca[[n]] [[j]],Range[k + 1 - n, k - 1 + n]], {j, k + 1 - n, k - 1 + n}], {n, 1, k}];
Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 10], {i, 1, stages - 1}]

Conjectures from Colin Barker, Apr 23 2017: (Start)
G.f.: (1 - x^2 + 10*x^4) / ((1 - x)*(1 - 10*x^2)).
a(n) = 10*(10^(n/2) - 1)/9 for n>1 and even.
a(n) = (10^((n+1)/2) - 10)/9 for n>1 and odd.
a(n) = a(n-1) + 10*a(n-2) - 10*a(n-3) for n>2.
(End)

A285613 Binary representation of the diagonal from the origin to the corner of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 62", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 10, 10, 100, 1100, 11000, 111000, 1110000, 11110000, 111100000, 1111100000, 11111000000, 111111000000, 1111110000000, 11111110000000, 111111100000000, 1111111100000000, 11111111000000000, 111111111000000000, 1111111110000000000, 11111111110000000000

Offset: 0

Robert Price, Apr 22 2017

Initialized with a single black (ON) cell at stage zero.

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Robert Price, Table of n, a(n) for n = 0..126
Robert Price, Diagrams of first 20 stages
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Wolfram Research, Wolfram Atlas of Simple Programs
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata

Cf. A285612, A056453, A233411.

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 62; stages = 128;
rule = IntegerDigits[code, 2, 10];
g = 2 * stages + 1; (* Maximum size of grid *)
a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
ca = a;
ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
PrependTo[ca, a];
(* Trim full grid to reflect growth by one cell at each stage *)
k = (Length[ca[[1]]] + 1)/2;
ca = Table[Table[Part[ca[[n]] [[j]],Range[k + 1 - n, k - 1 + n]], {j, k + 1 - n, k - 1 + n}], {n, 1, k}];
Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 10], {i, 1, stages - 1}]

Conjectures from Colin Barker, Apr 23 2017: (Start)
G.f.: (1 - 100*x^2 + 1000*x^4) / ((1 - 10*x)*(1 - 10*x^2)).
a(n) = (10^n - 10^(n/2)) / 9 for n>1 and even.
a(n) = (10^n - 10^(n/2+1/2)) / 9 for n>1 and odd.
a(n) = 10*a(n-1) + 10*a(n-2) - 100*a(n-3) for n>2.
(End)

A321434 Triangle read by rows; T(n,k) is the number of achiral rows of n colors using exactly k colors.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 6, 6, 0, 1, 6, 6, 0, 1, 14, 36, 24, 0, 1, 14, 36, 24, 0, 1, 30, 150, 240, 120, 0, 1, 30, 150, 240, 120, 0, 1, 62, 540, 1560, 1800, 720, 0, 1, 62, 540, 1560, 1800, 720, 0, 1, 126, 1806, 8400, 16800, 15120, 5040, 0, 1, 126, 1806, 8400, 16800, 15120, 5040

Offset: 0

Robert A. Russell, Nov 09 2018

Each zero in the data is the beginning of a new row.

Same as A131689, with rows (except for the first) repeated. - Joerg Arndt, Sep 08 2019

			The triangle begins with T(0,0):
1
0 1
0 1
0 1   2
0 1   2
0 1   6     6
0 1   6     6
0 1  14    36     24
0 1  14    36     24
0 1  30   150    240    120
0 1  30   150    240    120
0 1  62   540   1560   1800    720
0 1  62   540   1560   1800    720
0 1 126  1806   8400  16800   15120    5040
0 1 126  1806   8400  16800   15120    5040
0 1 254  5796  40824 126000  191520  141120   40320
0 1 254  5796  40824 126000  191520  141120   40320
0 1 510 18150 186480 834120 1905120 2328480 1451520 362880
For T(7,2)=14, the rows are AAABAAA, AABABAA, AABBBAA, ABAAABA, ABABABA, ABBABBA, ABBBBBA, BAAAAAB, BAABAAB, BABABAB, BABBBAB, BBAAABB, BBABABB, and BBBABBB.

Columns 0-6 are A000007, A057427, A056453, A056454, A056455, A056456, A056457.
Cf. A019538 (oriented), A305621 (unoriented), A305622 (chiral).
Cf. A131689.

Table[k! StirlingS2[Ceiling[n/2], k], {n, 0, 18}, {k, 0, (n+1)/2}] // Flatten

T(n,k) = k!*S2(ceiling(n/2),k), where S2 is the Stirling subset number A008277.

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A056463 Number of primitive (aperiodic) palindromes using exactly two different symbols.

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A285612 Binary representation of the diagonal from the corner to the origin of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 62", based on the 5-celled von Neumann neighborhood.

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A285613 Binary representation of the diagonal from the origin to the corner of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 62", based on the 5-celled von Neumann neighborhood.

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Mathematica

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A321434 Triangle read by rows; T(n,k) is the number of achiral rows of n colors using exactly k colors.

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