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.

A283508 Binary representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 643", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 10, 101, 1011, 10111, 101111, 1011111, 10111111, 101111111, 1011111111, 10111111111, 101111111111, 1011111111111, 10111111111111, 101111111111111, 1011111111111111, 10111111111111111, 101111111111111111, 1011111111111111111, 10111111111111111111
Offset: 0

Views

Author

Robert Price, Mar 09 2017

Keywords

Comments

Initialized with a single black (ON) cell at stage zero.
As far as the b-files reach (125 terms) this is the same as A267623. - R. J. Mathar, Mar 17 2017

References

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

Links

Crossrefs

Cf. A267623, A036563, A153893.

Programs

  • Mathematica
    CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 643; 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}]

Formula

Conjectures from Colin Barker, Mar 10 2017: (Start)
G.f.: (1 - x + x^2) / ((1 - x)*(1 - 10*x)).
a(n) = (91*10^n - 10) / 90 for n>0.
a(n) = 11*a(n-1) - 10*a(n-2) for n>2.
(End)
Equivalent conjecture: a(n) = A267623(n). - R. J. Mathar, Mar 17 2017

A306593 Least number k such that the determinant of the circulant matrix formed by its decimal digits is equal to n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 334, 65, 42, 76, 455, 41, 40, 98, 123, 667, 64, 52, 778, 788, 51, 50, 899, 63, 86, 7787, 2025885, 8788, 62, 74, 46996, 61, 60, 66898, 67997, 85, 73, 78998, 88899, 88999, 335, 72, 4579975, 878888, 71, 70, 10243, 5354, 355, 989999, 114
Offset: 0

Views

Author

Paolo P. Lava, Feb 27 2019

Keywords

Comments

Here only the least numbers are listed: e.g., a(75) = 1031, even if 10002110 also produces 75.
The sequence is infinite because any number of the form (91*10^n - 10) / 90 for n > 0 (A267623 or A283508) has the determinant of the circulant matrix equal to n but, in general, it is not the least possible term. - Giovanni Resta, Mar 06 2019

Examples

			                        | 3 3 4 |
a(10) = 334 because det | 4 3 3 | = 10
                        | 3 4 3 |
.
and 334 is the least number to have this property.
.
                          | 4 6 9 9 6 |
                          | 6 4 6 9 9 |
a(34) = 46996 because det | 9 6 4 6 9 | = 34
                          | 9 9 6 4 6 |
                          | 6 9 9 6 4 |
.
and 46996 is the least number to have this property.

Links

Crossrefs

Cf. A177894, A219324, A267623, A283508, A306662, A323485, A323486.

Programs

  • Maple
    with(linalg): P:=proc(q) local a,b,c,d,j,k,i,n,t;
print(0); for i from 1 to q do for n from 1 to q do
d:=ilog10(n)+1; a:=convert(n, base, 10); c:=[];
for k from 1 to nops(a) do c:=[op(c), a[-k]]; od; t:=[op([]), c];
for k from 2 to d do b:=[op([]), c[nops(c)]];
for j from 1 to nops(c)-1 do
b:=[op(b), c[j]]; od;  c:=b; t:=[op(t), c]; od;
if i=det(t) then print(n); break; fi; od; od; end: P(10^7);
  • PARI
    md(n) = my(d = if (n, digits(n), [0])); matdet(matrix(#d, #d, i, j, d[1+lift(Mod(j-i, #d))]));
a(n) = my(k=0); while(md(k) != n, k++); k; \\ Michel Marcus, Mar 20 2019

Formula

A177894(a(n)) = n when a(n) >= 0. - Rémy Sigrist, Feb 27 2019

A290112 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 643", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 11, 101, 1011, 10111, 101111, 1011111, 10111111, 101111111, 1011111111, 10111111111, 101111111111, 1011111111111, 10111111111111, 101111111111111, 1011111111111111, 10111111111111111, 101111111111111111, 1011111111111111111, 10111111111111111111
Offset: 0

Views

Author

Robert Price, Jul 19 2017

Keywords

Comments

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

References

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

Links

Crossrefs

Cf. A290111, A290113, A290114, A283508, A267623.

Programs

  • Mathematica
    CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 643; 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}]

Formula

From Chai Wah Wu, Apr 02 2024: (Start)
a(n) = 11*a(n-1) - 10*a(n-2) for n > 3.
G.f.: (10*x^3 - 10*x^2 + 1)/((x - 1)*(10*x - 1)). (End)
Showing 1-3 of 3 results.

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