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 1-5 of 5 results.

A080925 Binomial transform of Jacobsthal gap sequence (A080924).

Original entry on oeis.org

0, 1, 5, 13, 41, 121, 365, 1093, 3281, 9841, 29525, 88573, 265721, 797161, 2391485, 7174453, 21523361, 64570081, 193710245, 581130733, 1743392201, 5230176601, 15690529805, 47071589413, 141214768241, 423644304721, 1270932914165
Offset: 0

Views

Author

Paul Barry, Feb 26 2003

Keywords

Links

Crossrefs

Cf. A080926. Essentially the same as A046717.

Programs

  • Mathematica
    CoefficientList[Series[x (1 + 3 x) / ((1 + x) (1 - 3 x)), {x, 0, 40}], x] (* Vincenzo Librandi, Aug 05 2013 *)

Formula

a(n)=Sum{k=1..n, Binomial(n, 2k-2)2^(2k-2)}
a(n)=(3^n-2*0^n+(-1)^n)/2; G.f.: x(1+3x)/((1+x)(1-3x)); E.g.f.: (exp(3x)-2exp(0)+exp(-x))/2. - Paul Barry, May 19 2003

A080610 Partial sums of Jacobsthal gap sequence.

Original entry on oeis.org

0, 1, 4, 5, 20, 21, 84, 85, 340, 341, 1364, 1365, 5460, 5461, 21844, 21845, 87380, 87381, 349524, 349525, 1398100, 1398101, 5592404, 5592405, 22369620, 22369621, 89478484, 89478485, 357913940, 357913941, 1431655764, 1431655765, 5726623060
Offset: 0

Views

Author

Paul Barry, Feb 26 2003

Keywords

Links

Crossrefs

Cf. A075427, A080924, A094025.

Programs

  • Magma
    [2^n+(-2)^n/3-(-1)^n/2-5/6: n in [0..30]]; // Vincenzo Librandi, Aug 05 2013
  • Mathematica
    CoefficientList[Series[x (1 + 4 x) / ((1 - x^2) (1 - 4 x^2)), {x, 0, 40}], x] (* Vincenzo Librandi, Aug 05 2013 *)
LinearRecurrence[{0,5,0,-4},{0,1,4,5},40] (* Harvey P. Dale, Nov 11 2021 *)

Formula

a(2n-1) = A001045(2n) = A002450(n); a(2n) = A001045(2n) - 1 = A002450(n) - 1.
G.f.: x*(1+4*x)/((1-x^2)*(1-4x^2)). - Ralf Stephan, Sep 16 2003
a(n) = 2^n+(-2)^n/3-(-1)^n/2-5/6. - Paul Barry, Apr 22 2004
a(n) = a(n-1)*4 if n even; a(n) = a(n-1)+1 if n odd. - Philippe Deléham, Apr 22 2013

A285473 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 3", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 11, 1, 1111, 1, 111111, 1, 11111111, 1, 1111111111, 1, 111111111111, 1, 11111111111111, 1, 1111111111111111, 1, 111111111111111111, 1, 11111111111111111111, 1, 1111111111111111111111, 1, 111111111111111111111111, 1, 11111111111111111111111111, 1
Offset: 0

Views

Author

Robert Price, Apr 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. A285474, A080924, A285475.

Programs

  • Mathematica
    CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 3; 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, Apr 19 2017: (Start)
G.f.: (1 + 11*x - 100*x^2) / ((1 - x)*(1 + x)*(1 - 10*x)*(1 + 10*x)).
a(n) = (4 + 5*(-1)^n - (-2)^n*5^(1+n) + 2^n*5^(1+n))/9.
a(n) = 101*a(n-2) - 100*a(n-4) for n>3.
(End)

A285474 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 3", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 11, 100, 1111, 10000, 111111, 1000000, 11111111, 100000000, 1111111111, 10000000000, 111111111111, 1000000000000, 11111111111111, 100000000000000, 1111111111111111, 10000000000000000, 111111111111111111, 1000000000000000000, 11111111111111111111
Offset: 0

Views

Author

Robert Price, Apr 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. A285473, A080924, A285475.

Programs

  • Mathematica
    CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 3; 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, Apr 19 2017: (Start)
G.f.: (1 + 11*x - x^2) / ((1 - x)*(1 + x)*(1 - 10*x)*(1 + 10*x)).
a(n) = (-1 - (-10)^n + (-1)^n + 19*10^n)/18.
a(n) = 101*a(n-2) - 100*a(n-4) for n>3.
(End)

A285475 Decimal 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 3", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 3, 4, 15, 16, 63, 64, 255, 256, 1023, 1024, 4095, 4096, 16383, 16384, 65535, 65536, 262143, 262144, 1048575, 1048576, 4194303, 4194304, 16777215, 16777216, 67108863, 67108864, 268435455, 268435456, 1073741823, 1073741824, 4294967295, 4294967296
Offset: 0

Views

Author

Robert Price, Apr 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. A285473, A285474, A080924.
Cf. A083420.
Cf. A000302 (even bisection), A024036 (odd bisection).

Programs

  • Mathematica
    CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 3; 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 Colin Barker, Apr 19 2017: (Start)
G.f.: (1 + 3*x - x^2) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)).
a(n) = (-1 - (-2)^n + (-1)^n + 3*2^n)/2.
a(n) = 5*a(n-2) - 4*a(n-4) for n>3. (End)
a(2*n-1) + a(2*n) = A083420(n). - Paul Curtz, Dec 16 2024
Showing 1-5 of 5 results.

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