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.

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A135351 a(n) = (2^n + 3 - 7*(-1)^n + 3*0^n)/6; or a(0) = 0 and for n > 0, a(n) = A005578(n-1) - (-1)^n.

Original entry on oeis.org

0, 2, 0, 3, 2, 7, 10, 23, 42, 87, 170, 343, 682, 1367, 2730, 5463, 10922, 21847, 43690, 87383, 174762, 349527, 699050, 1398103, 2796202, 5592407, 11184810, 22369623, 44739242, 89478487, 178956970, 357913943, 715827882, 1431655767, 2863311530, 5726623063, 11453246122, 22906492247, 45812984490
Offset: 0

Views

Author

Miklos Kristof, Dec 07 2007

Keywords

Comments

Partial sums of A155980 for n > 2. - Klaus Purath, Jan 30 2021

Links

Crossrefs

Cf. A005578, A099754.
Cf. A001045, A327767, A328284.
Cf. A007583, A062092, A087289, A020988 (even bisection), A163834 (odd bisection), A078008, A084247, A181565.

Programs

  • GAP
    List([0..40], n-> (2^n+3-7*(-1)^n+3*0^n)/6); # G. C. Greubel, Sep 02 2019
  • Magma
    a135351:=func< n | (2^n+3-7*(-1)^n+3*0^n)/6 >; [ a135351(n): n in [0..32] ]; // Klaus Brockhaus, Dec 05 2009
  • Maple
    G(x):=x*(2 - 4*x + x^2)/((1-x^2)*(1-2*x)): f[0]:=G(x): for n from 1 to 30 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n]/n!,n=0..30);
  • Mathematica
    Join[{0}, Table[(2^n +3 -7*(-1)^n)/6, {n,40}]] (* G. C. Greubel, Oct 11 2016 *)
LinearRecurrence[{2,1,-2},{0,2,0,3},40] (* Harvey P. Dale, Feb 13 2024 *)
  • PARI
    a(n) = (2^n + 3 - 7*(-1)^n + 3*0^n)/6; \\ Michel Marcus, Oct 11 2016
  • Sage
    [(2^n+3-7*(-1)^n+3*0^n)/6 for n in (0..40)] # G. C. Greubel, Sep 02 2019

Formula

G.f.: x*(2 - 4*x + x^2)/((1-x^2)*(1-2*x)).
E.g.f.: (exp(2*x) + 3*exp(x) - 7*exp(-x) + 3)/6.
From Paul Curtz, Dec 20 2020: (Start)
a(n) + (period 2 sequence: repeat [1, -2]) = A328284(n+2).
a(n+1) + (period 2 sequence: repeat [-2, 1]) = A001045(n).
a(n+1) + (period 2 sequence: repeat [-1, 0]) = A078008(n).
a(n+1) + (period 2 sequence : repeat [-3, 2]) = -(-1)^n*A084247(n).
a(n+4) = a(n+1) + 7*A001045(n).
a(n+4) + a(n+1) = A181565(n).
a(2*n+2) + a(2*n+3) = A087289(n) = 3*A007583(n).
a(2*n+1) = A163834(n), a(2*n+2) = A020988(n). (End)

Extensions

First part of definition corrected by Klaus Brockhaus, Dec 05 2009

A283642 Decimal 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 678", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 3, 5, 11, 21, 43, 85, 171, 341, 683, 1365, 2731, 5461, 10923, 21845, 43691, 87381, 174763, 349525, 699051, 1398101, 2796203, 5592405, 11184811, 22369621, 44739243, 89478485, 178956971, 357913941, 715827883, 1431655765, 2863311531, 5726623061, 11453246123
Offset: 0

Views

Author

Robert Price, Mar 12 2017

Keywords

Comments

Initialized with a single black (ON) cell at stage zero.
Similar to A001045.
It is not difficult to prove that one has indeed a(n) = round(4*2^n/3) = A001045(n+2) for all n. The proof as well as the growth of the pattern is nearly identical to that of the toothpick sequence A139250. - M. F. Hasler, Feb 13 2020
The decimal representations of the n-th interval of elementary cellular automata rules 28 and 156 (see A266502 and A266508) generate this sequence. - Karl V. Keller, Jr., Sep 03 2021

Links

Crossrefs

Cf. A283641, A266502, A266508, A086893, A001045.

Programs

  • Mathematica
    CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 678; 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]], 2], {i ,1, stages - 1}]
  • Python
    print([(4*2**n + 1)//3 for n in range(50)]) # Karl V. Keller, Jr., Sep 03 2021

Formula

From Colin Barker, Mar 14 2017: (Start)
G.f.: (1 + 2*x) / ((1 + x)*(1 - 2*x)).
a(n) = (2^(n+2) - 1) / 3 for n even.
a(n) = (2^(n+2) + 1) / 3 for n odd.
a(n) = a(n-1) + 2*a(n-2) for n>1.
(End)
I.e., a(n) = A001045(n+2) = A154917(n+2) = A167167(n+2) = |A077925(n+1)| = A328284(n+5) = round(4*2^n/3), cf. comments. - M. F. Hasler, Feb 13 2020
E.g.f.: (4*exp(2*x) - exp(-x))/3. - Stefano Spezia, Feb 13 2020
a(n) = floor((4*2^n + 1)/3). - Karl V. Keller, Jr., Sep 03 2021
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