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 1-6 of 6 results.

A253767 Partial sums of A247666.

Original entry on oeis.org

1, 8, 15, 40, 47, 96, 121, 224, 231, 280, 329, 504, 529, 704, 807, 1216, 1223, 1272, 1321, 1496, 1545, 1888, 2063, 2784, 2809, 2984, 3159, 3784, 3887, 4608, 5017, 6656, 6663, 6712, 6761, 6936, 6985, 7328, 7503, 8224, 8273, 8616, 8959, 10184, 10359, 11584, 12305, 15168, 15193, 15368, 15543, 16168
Offset: 0

Views

Author

Omar E. Pol, Jan 29 2015

Keywords

Comments

Also, total number of ON cells after n generations in a three-dimensional cellular automaton where A247666(n) gives the number of ON cells in the n-th level of the structure starting from the top. An ON cell remains ON forever. An ON cell is an hexagonal prism of height 1. We start with a single ON cell. The structure looks like an irregular stepped pyramid, apparently with a like-hexagonal base.

Links

Crossrefs

Cf. A116520, A245542, A247666, A253908.

A102900 a(n) = 3*a(n-1) + 4*a(n-2), a(0)=a(1)=1.

Original entry on oeis.org

1, 1, 7, 25, 103, 409, 1639, 6553, 26215, 104857, 419431, 1677721, 6710887, 26843545, 107374183, 429496729, 1717986919, 6871947673, 27487790695, 109951162777, 439804651111, 1759218604441, 7036874417767, 28147497671065
Offset: 0

Views

Author

Paul Barry, Jan 17 2005

Keywords

Comments

Binomial transform of A102901.
Hankel transform is = 1,6,0,0,0,0,0,0,0,0,0,0,... - Philippe Deléham, Nov 02 2008

References

  • Maria Paola Bonacina and Nachum Dershowitz, Canonical Inference for Implicational Systems, in Automated Reasoning, Lecture Notes in Computer Science, Volume 5195/2008, Springer-Verlag.

Links

Crossrefs

Cf. A001045, A004171, A046717, A086901, A102901, A247666 (which appears to be the run length transform of this sequence).

Programs

  • Haskell
    a102900 n = a102900_list !! n
a102900_list = 1 : 1 : zipWith (+)
               (map (* 4) a102900_list) (map (* 3) $ tail a102900_list)
-- Reinhard Zumkeller, Feb 13 2015
  • Magma
    [n le 2 select 1 else 3*Self(n-1)+4*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Dec 28 2015
  • Mathematica
    a[n_]:=(MatrixPower[{{2,2},{3,1}},n].{{2},{1}})[[2,1]]; Table[a[n],{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *)
LinearRecurrence[{3, 4}, {1, 1}, 30] (* Vincenzo Librandi, Dec 28 2015 *)
  • PARI
    a(n)=([0,1; 4,3]^n*[1;1])[1,1] \\ Charles R Greathouse IV, Mar 28 2016
  • SageMath
    A102900=BinaryRecurrenceSequence(3,4,1,1)
[A102900(n) for n in range(51)] # G. C. Greubel, Dec 09 2022

Formula

G.f.: (1-2*x)/(1-3*x-4*x^2).
a(n) = (2*4^n + 3*(-1)^n)/5.
a(n) = ceiling(4^n/5) + floor(4^n/5) = (ceiling(4^n/5))^2 - (floor(4^n/5))^2.
a(n) + a(n+1) = 2^(2*n+1) = A004171(n).
a(n) = Sum_{k=0..n} binomial(2*n-k, 2*k)*2^k. - Paul Barry, Jan 20 2005
a(n) = upper left term in the 2 X 2 matrix [1,3; 2,2]^n. - Gary W. Adamson, Mar 14 2008
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(8*4^k-3*(-1)^k)/(x*(8*4^k-3*(-1)^k) + (2*4^k+3*(-1)^k)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 28 2013
a(n) = 2^(2*n-1) - a(n-1), a(1)=1. - Ben Paul Thurston, Dec 27 2015; corrected by Klaus Purath, Aug 02 2020
From Klaus Purath, Aug 02 2020: (Start)
a(n) = 4*a(n-1) + 3*(-1)^n.
a(n) = 6*4^(n-2) + a(n-2), n>=2. (End)

A247640 Number of ON cells after n generations of "Odd-Rule" cellular automaton on hexagonal lattice based on 6-celled neighborhood.

Original entry on oeis.org

1, 6, 6, 24, 6, 36, 24, 96, 6, 36, 36, 144, 24, 144, 96, 384, 6, 36, 36, 144, 36, 216, 144, 576, 24, 144, 144, 576, 96, 576, 384, 1536, 6, 36, 36, 144, 36, 216, 144, 576, 36, 216, 216, 864, 144, 864, 576, 2304, 24, 144, 144, 576, 144, 864
Offset: 0

Views

Author

N. J. A. Sloane, Sep 22 2014

Keywords

Comments

The neighborhood of a cell consists of the six surrounding cells (but not the cell itself). A cell is ON at generation n iff an odd number of its neighbors were ON at the previous generation. We start with one ON cell.
This is the Run Length Transform of the sequence 1, 6, 24, 96, 384, 1536, 6144, 24576, ... (almost certainly A164908, or 1 followed by A002023).
It appears that this is also the sequence corresponding to the odd-rule cellular automaton defined by OddRule 356 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link). - N. J. A. Sloane, Feb 26 2015

Links

Crossrefs

Cf. A164908, A001023, A071053, A160239, A247666.

Programs

  • Maple
    C := f->`if`(type(f,`+`),nops(f),1);
f := 1/x+x+1/y+y+1/(x*y)+x*y;
g := n->expand(f^n) mod 2;
[seq(C(g(n)),n=0..100)];
  • Mathematica
    A247640[n_] := Total[CellularAutomaton[{42, {2, {{1, 1, 0}, {1, 0, 1}, {0, 1, 1}}}, {1, 1}}, {{{1}}, 0}, {{{n}}}], 2]; Array[A247640, 54, 0] (* JungHwan Min, Sep 06 2016 *)
A247640L[n_] := Total[#, 2] & /@ CellularAutomaton[{42, {2, {{1, 1, 0}, {1, 0, 1}, {0, 1, 1}}}, {1, 1}}, {{{1}}, 0}, n]; A247640L[53] (* JungHwan Min, Sep 06 2016 *)

Formula

a(n) = number of terms in expansion of f^n mod 2, where f = 1/x+x+1/y+y+1/(x*y)+x*y (mod 2);

A275667 Number of ON cells after n generations in a 2-dimensional "Odd-Rule" cellular automaton on triangular tiling.

Original entry on oeis.org

1, 3, 7, 9, 7, 21, 25, 27, 7, 21, 49, 63, 25, 75, 103, 81, 7, 21, 49, 63, 49, 147, 175, 189, 25, 75, 175, 225, 103, 309, 409, 243, 7, 21, 49, 63, 49, 147, 175, 189, 49, 147, 343, 441, 175, 525, 721, 567, 25, 75, 175, 225, 175, 525, 625, 675, 103, 309, 721
Offset: 0

Views

Author

Kovba Alexey, Aug 04 2016

Keywords

Comments

Each triangular tile has 3 neighbors. A cell is ON in a given generation if and only if there was an odd number of ON cells among the three nearest neighbors in the preceding generation.
At the initial moment there is a single ON cell.
Given pattern replicates after a number of generations which is a power of 2 when a(n) = 7.
Number of cells on each even step minus one is divisible by 6.
By analogy with the Ekhad, Sloane, Zeilberger link, one may suppose that using ternary expansion of n, recurrence relations for a(n) can be obtained and proved.
From Andrey Zabolotskiy, Aug 04 2016: (Start)
If the first conjecture from the Formula section is true then the fact that the right border of the triangle (see Example) gives A000244 follows directly from it.
If the second conjecture is true then the numbers just before the right border give A102900.
Since the 7 cells which are ON at the beginning of every row are farther and farther away from each other, the n-th term of a row (with offset 0) is a(n)*7 for not very large n.
See also comments to A247666.
(End)
This is ETA rule 170. See the Sadat-Benedek reference for proof of pattern replication. - Paul Cousin, Apr 22 2025

Examples

			From _Omar E. Pol_, Aug 04 2016: (Start)
Written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:
1;
3;
7, 9;
7, 21, 25, 27;
7, 21, 49, 63, 25, 75, 103, 81;
7, 21, 49, 63, 49, 147, 175, 189, 25, 75, 175, 225, 103, 309, 409, 243;
...
It appears that the right border gives A000244.
(End)

Links

Crossrefs

Cf. A160239 (square tiling analog), A247640, A247666 (hexagonal tiling analogs).
Cf. A000244, A011782, A102900, A382971.
Pattern replicating ETA rules: A383369 (rule 90).

Formula

a(0) = 1. Conjecture: a(2*t+1) = 3*a(t).
Conjectures: a(8*t+6) = 3*a(4*t+2) + 4*a(2*t), a(8*t+2) = 3*a(4*t) + 4*a(2*t), a(4*t) = a(2*t). These conjectured formulas together give recurrent relations for a(n) for any n. Also, obviously a(2*n) = A247666(n). - Andrey Zabolotskiy, Aug 04 2016

A335794 Number of ON cells at n-th generation in an "Ulam-Warburton and Friedkin Replicator" hybrid two-dimensional cellular automaton.

Original entry on oeis.org

0, 1, 7, 13, 31, 37, 67, 85, 139, 145, 175, 205, 283, 301, 391, 445, 595, 601, 631, 661, 739, 769, 895, 973, 1195, 1213, 1303, 1393, 1627, 1681, 1927, 2077, 2467, 2473, 2503, 2533, 2611, 2641, 2767, 2845, 3067, 3097, 3223, 3349, 3667, 3745, 4111, 4333, 4939, 4957
Offset: 0

Views

Author

Cody B Duncan, Jun 23 2020

Keywords

Comments

This 2D CA uses the neighborhood:
[0 X X]
[X X X]
[X X 0]
If a cell has an even number of ON neighbors and it is currently OFF, stay OFF; otherwise turn ON.
The results are similar to those for A151723, but with a distorted grid.
(The "look" keyword refers to the animation. - N. J. A. Sloane, Jul 03 2020)

References

  • S. M. Ulam, On some mathematical problems connected with patterns of growth of figures, pp. 215-224 of R. E. Bellman, ed., Mathematical Problems in the Biological Sciences, Proc. Sympos. Applied Math., Vol. 14, Amer. Math. Soc., 1962 (see Example 6, page 224).

Links

Crossrefs

Cf. A102900, A147562, A151723, A247666, A335795.

A335796 Number of ON cells at n-th generation in the "Ulam-Warburton and Friedkin Replicator" hybrid two-dimensional cellular automaton.

Original entry on oeis.org

0, 1, 9, 17, 33, 41, 97, 129, 193, 209, 265, 313, 433, 481, 609, 705, 849, 889, 977, 1073, 1241, 1369, 1641, 1769, 2025, 2089, 2225, 2337, 2601, 2793, 3113, 3337, 3705, 3785, 3921, 4081, 4361, 4601, 5033, 5273, 5641, 5849, 6193, 6449, 6937, 7161, 7641, 7929, 8489, 8625, 8857, 9145
Offset: 0

Views

Author

Cody B Duncan, Jun 24 2020

Keywords

Comments

The 2D CA uses the Moore neighborhood (all eight orthogonally and diagonally adjacent cells are in the neighborhood).
Generation follows the following algorithm: If a cell has an even number of ON neighbors and it is currently OFF, stay OFF; otherwise turn ON.

Crossrefs

Cf. A147562, A247666, A335794
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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