A265688 -id:A265688 - cp's OEIS frontend

A037576 Base-4 digits are, in order, the first n terms of the periodic sequence with initial period 1,3.

1, 7, 29, 119, 477, 1911, 7645, 30583, 122333, 489335, 1957341, 7829367, 31317469, 125269879, 501079517, 2004318071, 8017272285, 32069089143, 128276356573, 513105426295, 2052421705181, 8209686820727, 32838747282909

Offset: 1

Clark Kimberling

Partial sums of A255465. - Klaus Purath, Mar 18 2021

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Rule 190
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata
Index entries for linear recurrences with constant coefficients, signature (4,1,-4).

Cf. A007090 (numbers in base 4), A037582 (decimal), A265688 (binary), A118111.

I:=[1, 7, 29]; [n le 3 select I[n] else 4*Self(n-1)+Self(n-2)-4*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 22 2012

CoefficientList[Series[(1+3*x)/((x-1)*(4*x-1)*(1+x)),{x,0,30}],x] (*or*) LinearRecurrence[{4,1,-4},{1,7,29},40] (* Vincenzo Librandi, Jun 22 2012 *)

my(x='x+O('x^99)); Vec(x*(1+3*x)/((1-x)*(1-4*x)*(1+x))) \\ Altug Alkan, Sep 21 2018

print([7*4**n//15 for n in range(1,30)]) # Karl V. Keller, Jr., Mar 09 2021

G.f.: x*(1+3*x)/((1-x)*(1-4*x)*(1+x)). - Vincenzo Librandi, Jun 22 2012
a(n) = 4*a(n-1) + a(n-2) - 4*a(n-3). - Vincenzo Librandi, Jun 22 2012
a(n) = (7*4^n + 3*(-1)^n - 10)/15. - Bruno Berselli, Jun 22 2012, corrected by Klaus Purath, Mar 18 2021.
a(n) = floor(7*4^n/15). - Karl V. Keller, Jr., Mar 09 2021
From Klaus Purath, Mar 18 2021: (Start)
a(n) = 16*a(n-2) - 3*(-1)^n + 10, assuming that a(0) = 0.
a(n) = 4*a(n-1) + 2 + (-1)^n.
a(n) = 5*a(n-1) - 4*a(n-2) + 2*(-1)^n, n > 2. (End)

A118111 Binary representation of n-th iteration of the Rule 190 elementary cellular automaton starting with a single black cell.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1

Offset: 0

Eric W. Weisstein, Apr 13 2006

Row n has length 2*n+1. - Hans Havermann, May 26 2002

			From _Michael De Vlieger_, Aug 21 2020: (Start)
Irregular array begins:
0:                             1
1:                          1  1  1
2:                       1  1  1  0  1
3:                    1  1  1  0  1  1  1
4:                 1  1  1  0  1  1  1  0  1
5:              1  1  1  0  1  1  1  0  1  1  1
6:           1  1  1  0  1  1  1  0  1  1  1  0  1
7:        1  1  1  0  1  1  1  0  1  1  1  0  1  1  1
8:     1  1  1  0  1  1  1  0  1  1  1  0  1  1  1  0  1
9:  1  1  1  0  1  1  1  0  1  1  1  0  1  1  1  0  1  1  1
... (End)

Robert Price, Table of n, a(n) for n = 0..9999
Eric Weisstein's World of Mathematics, Rule 190
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Cf. A265688 (binary rows), A037576 (decimal rows), A032766 (num 1's).

With[{nn = 9}, MapIndexed[#1[[#2 + 1 ;; 2 nn - #2 + 1]] & @@ {#1, nn - First[#2] + 1} &, CellularAutomaton[190, {{1}, 0}, nn]]] // Flatten (* Michael De Vlieger, Aug 21 2020 *)

cp's OEIS Frontend

A037576 Base-4 digits are, in order, the first n terms of the periodic sequence with initial period 1,3.

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