A037576 -id:A037576 - cp's OEIS frontend

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

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 *)

A265688 Binary representation of the n-th iteration of the "Rule 190" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 111, 11101, 1110111, 111011101, 11101110111, 1110111011101, 111011101110111, 11101110111011101, 1110111011101110111, 111011101110111011101, 11101110111011101110111, 1110111011101110111011101, 111011101110111011101110111, 11101110111011101110111011101

Offset: 0

Robert Price, Dec 13 2015

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Robert Price, Table of n, a(n) for n = 0..499
Eric Weisstein's World of Mathematics, Rule 190
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata
Index entries for linear recurrences with constant coefficients, signature (100,1,-100).

Cf. A037576 (decimal), A118111.

rule = 190; rows = 20; Table[FromDigits[Table[Take[CellularAutomaton[rule,{{1},0}, rows-1, {All,All}][[k]], {rows-k+1, rows+k-1}], {k,1,rows}][[k]]], {k,1,rows}]

print([11100*100**n//9999 for n in range(30)]) # Karl V. Keller, Jr., Aug 10 2021

From Colin Barker, Dec 13 2015 and Apr 18 2019: (Start)
a(n) = (-165*(-1)^n+37*100^(n+1)-202)/3333.
a(n) = (37*100^(n+1)-367)/3333 for n even.
a(n) = (37*100^(n+1)-37)/3333 for n odd.
a(n) = 100*a(n-1) + a(n-2) - 100*a(n-3) for n>2.
G.f.: (1+11*x) / ((1-x)*(1+x)*(1-100*x)).
(End)
a(n) = floor(11100*100^n/9999). - Karl V. Keller, Jr., Aug 10 2021

A028894 a(n) = either 4a(n-1)+1 or 4a(n-1)+3 depending on corresponding term of A005614, +1 for 0, +3 for 1.

Original entry on oeis.org

1, 5, 23, 95, 381, 1527, 6109, 24439, 97759, 391037, 1564151, 6256607, 25026429, 100105719, 400422877, 1601691511, 6406766047, 25627064189, 102508256759, 410033027037, 1640132108151, 6560528432607, 26242113730429

Offset: 1

John McNamara, Jan 12 2002

nonn

Related to A005614, A002450, A037576, A066744.

f[1]={0}; f[2]={1}; f[n_] := f[n]=Join[f[n-1], f[n-2]]; a[1]=1; a[n_] := a[n]=4a[n-1]+1+2f[9][[n]]; a/@Range[1, 30]

Edited by Dean Hickerson, Jan 14 2002

A066744 a(n) = either 4a(n-1)+1 or 4a(n-1)+3 depending on corresponding term of A005614, +3 for 0, +1 for 1.

Original entry on oeis.org

1, 7, 29, 117, 471, 1885, 7543, 30173, 120693, 482775, 1931101, 7724405, 30897623, 123590493, 494361975, 1977447901, 7909791605, 31639166423, 126556665693, 506226662775, 2024906651101, 8099626604405, 32398506417623

Offset: 1

John McNamara, Jan 16 2002

nonn

Ratio to terms of A028894 tends to 1.23459972586...

Related to A005614, A002450, A037576, A028894.

f[1]={0}; f[2]={1}; f[n_] := f[n]=Join[f[n-1], f[n-2]]; a[1]=1; a[n_] := a[n]=4a[n-1]+3-2f[9][[n]]; a/@Range[1, 30]

A102865 Base-4 digits are, in order, the first n terms of the sequence (1, 3, 21, 203, 2021, 20203, 202021, 2020203, 20202021, 202020203, ... ).

Original entry on oeis.org

1, 3, 9, 35, 137, 547, 2185, 8739, 34953, 139811, 559241, 2236963, 8947849, 35791395, 143165577, 572662307, 2290649225, 9162596899, 36650387593, 146601550371, 586406201481, 2345624805923, 9382499223689, 37529996894755, 150119987579017, 600479950316067

Offset: 0

Creighton Dement, Mar 01 2005

Index entries for linear recurrences with constant coefficients, signature (4,1,-4).

Cf. A037576.

FromDigits[IntegerDigits[#],4]&/@(NestList[FromDigits[Flatten[ IntegerDigits[#]/.{3->{2,1},1->{0,3}}]]&,1,30]) (* or *) LinearRecurrence[{4,1,-4},{1,3,9},31](* Harvey P. Dale, Mar 23 2012 *)

4^n = a(n) + A037576(n) for n >= 1.
a(n) + a(n+1) = A039301(n+2).
a(n) = 4*a(n-1) + a(n-2) - 4*a(n-3). - Harvey P. Dale, Mar 23 2012
G.f.: 1 + x*(3-3*x-4*x^2)/((1-x)*(1+x)*(1-4*x)). - Colin Barker, Aug 28 2012

More terms from Harvey P. Dale, Mar 23 2012

A140320 a(n) = A137576((3^n-1)/2).

Original entry on oeis.org

1, 3, 13, 55, 217, 811, 2917, 10207, 34993, 118099, 393661, 1299079, 4251529, 13817467, 44641045, 143489071, 459165025, 1463588515, 4649045869, 14721978583, 46490458681, 146444944843, 460255540933, 1443528742015, 4518872583697, 14121476824051, 44059007691037, 137260754729767

Offset: 0

Vladimir Shevelev, May 26 2008

nonn

Conjecture. a(n) = 2n*3^(n-1)+1.
If conjecture is true then limsup(A137576(n)/n)=infinity while liminf(A137576(n)/n)=2 with a realization on primes.
a(n) is also the number of edges in the graph generated from the n-dimensional hypercube (plus 1) in the following manner: connect all (d + 1)-dimensional faces to the d faces that are incident. Each d-dimensional face should be incident on (n - d) (d + 1)-dimensional faces. [Roy Liu (royliu(AT)cs.ucsd.edu), Jul 26 2010]

Wikipedia, Hypercube

Cf. A037576, A002326, A006694, A025192.

a137576(n) = my(t); sumdiv(2*n+1, d, eulerphi(d)/(t=znorder(Mod(2, d))))*t-t+1;
a(n) = a137576((3^n-1)/2); \\ Michel Marcus, Dec 18 2018

Sum_{m = 0}^{n} 2^(n - m) * binomial(n,m) is the number of m-dimensional faces in the n-dimensional hypercube. Consequently, Sum_{m = 0..n} (n - m) * 2^(n - m) * binomial(n,m) gives the number of incidence edges, which yields said sequence minus 1. The recurrence relation is: a(n) = 3 * a(n - 1) + 2 * 3^(n - 1) - 2. [Roy Liu (royliu(AT)cs.ucsd.edu), Jul 26 2010]

Empirical G.f.: (1-4*x+7*x^2)/(1-7*x+15*x^2-9*x^3). [Colin Barker, Jan 09 2012]

More terms from Michel Marcus, Dec 18 2018

cp's OEIS Frontend

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

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A265688 Binary representation of the n-th iteration of the "Rule 190" elementary cellular automaton starting with a single ON (black) cell.

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A028894 a(n) = either 4a(n-1)+1 or 4a(n-1)+3 depending on corresponding term of A005614, +1 for 0, +3 for 1.

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A066744 a(n) = either 4a(n-1)+1 or 4a(n-1)+3 depending on corresponding term of A005614, +3 for 0, +1 for 1.

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A102865 Base-4 digits are, in order, the first n terms of the sequence (1, 3, 21, 203, 2021, 20203, 202021, 2020203, 20202021, 202020203, ... ).

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A140320 a(n) = A137576((3^n-1)/2).

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