A110594 -id:A110594 - cp's OEIS frontend

A164346 a(n) = 3 * 4^n.

3, 12, 48, 192, 768, 3072, 12288, 49152, 196608, 786432, 3145728, 12582912, 50331648, 201326592, 805306368, 3221225472, 12884901888, 51539607552, 206158430208, 824633720832, 3298534883328, 13194139533312, 52776558133248, 211106232532992, 844424930131968

Offset: 0

Klaus Brockhaus, Aug 13 2009

Binomial transform of A000244 without initial 1.
Second binomial transform of A007283.
Third binomial transform of A010701.
Inverse binomial transform of A005053 without initial 1.
First differences of A024036. - Omar E. Pol, Feb 16 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (4).

Cf. A000302 (powers of 4), A000244 (powers of 3), A007283 (3*2^n), A010701 (all 3's), A005053, A002001, A096045, A140660 (3*4^n+1), A002023 (6*4^n), A002063(9*4^n), A056120, A084509.

Magma
```
[ 3*4^n: n in [0..22] ];
```

3 4^Range[0,30]  (* Harvey P. Dale, Mar 11 2011 *)

A164346(n)=3*4^n \\ M. F. Hasler, Jul 28 2015

def A164346(n): return 3<<(n<<1) # Chai Wah Wu, Aug 30 2024

a(n) = 4*a(n-1) for n > 1; a(0) = 3.
G.f.: 3/(1-4*x).
a(n) = A002001(n+1). a(n) = A096045(n)+2. a(n) = A140660(n)-1.
a(n) = A002023(n)/2. a(n) = A002063(n)/3. a(n) = A056120(n+3)/9.
Apparently a(n) = A084509(n+3)/2.
a(n) = A110594(n+1), n>1. - R. J. Mathar, Aug 17 2009
a(n) = 3*A000302(n). - Omar E. Pol, Feb 18 2013
a(n) = A000079(2*n) + A000079(2*n+1). - M. F. Hasler, Jul 28 2015
E.g.f.: 3*exp(4*x). - G. C. Greubel, Sep 15 2017

A110591 Number of digits in base-4 representation of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 0

Jonathan Vos Post, Jul 29 2005

Number of digits in A007090(n).

In terms of the repetition convolution operator #, where (sequence A) # (sequence B) = the sequence consisting of A(n) copies of B(n), this sequence is the repetition convolution A110594 # n. Over the set of positive infinite integer sequences, # gives a nonassociative noncommutative groupoid (magma) with a left identity (A000012) but no right identity, where the left identity is also a right nullifier and idempotent. For any positive integer constant c, the sequence c*A000012 = (c,c,c,c,...) is also a right nullifier; for c = 1, this is A000012; for c = 3 this is A010701.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A000012, A007090, A010701, A049804, A081604, A110594.

import Data.List (unfoldr)
a110591 0 = 1
a110591 n = length $
   unfoldr (\x -> if x == 0 then Nothing else Just (x, x `div` 4)) n
-- Reinhard Zumkeller, Apr 22 2011

A110592 := proc(n)
    if n = 0 then
        1;
    else
        1+floor(log[4](n)) ;
    end if;
end proc:
seq(A110592(n),n=0..50) ; # R. J. Mathar, Sep 02 2020

a[n_] := If[n == 0, 1, Floor[Log[4, n]] + 1];
a /@ Range[0, 100] (* Jean-François Alcover, Nov 24 2020 *)

G.f.: 1 + (1/(1 - x))*Sum_{k>=0} x^(4^k). - Ilya Gutkovskiy, Jan 08 2017

a(n) = floor(log_4(n)) + 1 for n >= 1. - Petros Hadjicostas, Dec 12 2019

A092898 Expansion of (1 - 4x + 4x^2 - 4x^3)/(1 - 4x).

Original entry on oeis.org

1, 0, 4, 12, 48, 192, 768, 3072, 12288, 49152, 196608, 786432, 3145728, 12582912, 50331648, 201326592, 805306368, 3221225472, 12884901888, 51539607552, 206158430208, 824633720832, 3298534883328, 13194139533312, 52776558133248

Offset: 0

Paul Barry, Mar 12 2004

Partial sums are A092896.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4).

Cf. A002001, A092896, A110594.

[1,0,4] cat [3*4^(n-2): n in [3..30]]; // G. C. Greubel, Feb 21 2021

a:= n-> 3*4^n/16+13*0^n/16+add(binomial(n,k)*(-1)^k*(3*k/4+k*(k-1)/2), k=0..n):
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 03 2018

Join[{1, 0, 4}, LinearRecurrence[{4}, {12}, 22]] (* Jean-François Alcover, Sep 16 2019 *)

Vec((1 -4*x +4*x^2 -4*x^3)/(1-4*x) + O(x^30)) \\ Andrew Howroyd, Nov 03 2018

[1,0,4]+[3*4^(n-2) for n in (3..30)] # G. C. Greubel, Feb 21 2021

a(n+2) = 4 * A002001(n).
a(n) = (3*4^n + 13*0^n)/16 + Sum_{k=0..n} binomial(n, k)*(-1)^k*(3*k/4 + k*(k-1)/2).
G.f.: 1 - x + 8*x^2 + 2*x/G(0), where G(k) = 1 + 1/(1 - x*(3*k+4)/(x*(3*k+7) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 11 2013
a(n) = A110594(n-1) for n >= 2. - Georg Fischer, Nov 03 2018
From G. C. Greubel, Feb 21 2021: (Start)
a(n) = (3*4^n +16*[n=2] -12*[n=1] +13*0^n)/16.
E.g.f.: (13 -12*x + 8*x^2 + 3*exp(4*x))/16. (End)

A110595 a(1)=5. For n > 1, a(n) = 4*5^(n-1) = A005054(n).

Original entry on oeis.org

5, 20, 100, 500, 2500, 12500, 62500, 312500, 1562500, 7812500, 39062500, 195312500, 976562500, 4882812500, 24414062500, 122070312500, 610351562500, 3051757812500, 15258789062500, 76293945312500, 381469726562500

Offset: 1

Jonathan Vos Post, Jul 29 2005

a(n) is the number of n-digit integers that contain only even digits (A014263). - Bernard Schott, Nov 11 2022

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5).

Cf. A000351, A007091, A014263, A081604, A110591, A110592, A110593, A110594.

Join[{5},NestList[5#&,20,20]] (* Harvey P. Dale, Jun 19 2013 *)
Rest[CoefficientList[Series[5 x (1 - x)/(1 - 5 x), {x,0,50}], x]] (* G. C. Greubel, Sep 01 2017 *)

my(x='x+O('x^50)); Vec(5*x*(1-x)/(1-5*x)) \\ G. C. Greubel, Sep 01 2017

O.g.f.: 5*x*(1-x)/(1-5*x). - Better definition from R. J. Mathar, May 13 2008

Sum_{n>=1} 1/a(n) = 21/80. - Bernard Schott, Nov 11 2022

Better definition from R. J. Mathar, May 13 2008

Incorrect comment removed by Michel Marcus, Nov 11 2022

A383369 Population of elementary triangular automaton rule 90 at generation n, starting from a lone 1 cell at generation 0.

Original entry on oeis.org

1, 4, 6, 12, 6, 24, 24, 48, 6, 24, 36, 72, 24, 96, 96, 192, 6, 24, 36, 72, 36, 144, 144, 288, 24, 96, 144, 288, 96, 384, 384, 768, 6, 24, 36, 72, 36, 144, 144, 288, 36, 144, 216, 432, 144, 576, 576, 1152, 24, 96, 144, 288, 144, 576, 576, 1152, 96, 384, 576, 1152, 384, 1536, 1536, 3072, 6

Offset: 0

Paul Cousin, Apr 24 2025

nonn

An Elementary Triangular Automaton (ETA) is a cellular automaton in the triangular grid where cells hold binary states and rules are local to the first neighborhood. There are 256 possible ETA rules.
Rule 90 (1011010 in binary):
  -----------------------------------------------
  |state of the cell            |1|1|1|1|0|0|0|0|
  |sum of the neighbors' states |3|2|1|0|3|2|1|0|
  |cell's next state            |0|1|0|1|1|0|1|0|
  -----------------------------------------------
This is one of the 4 ETA rules (85, 90, 165 and 170) that replicates the pattern given as initial condition.

			Written as an irregular triangle with row lengths A000079, starting from n=1, the sequence begins:
  4;
  6, 12;
  6, 24, 24, 48;
  6, 24, 36, 72, 24, 96, 96, 192;
  6, 24, 36, 72, 36, 144, 144, 288, 24, 96, 144, 288, 96, 384, 384, 768;
...
It appears that the right border gives A110594.

Paul Cousin, Table of n, a(n) for n = 0..16384
Paul Cousin, Illustration for n = 0..128
Paul Cousin, Triangular Automata
Paul Cousin, Rule 90
Paul Cousin, Triangular Automata Integer Sequences
Paul Cousin, Triangular Automata: The 256 Elementary Cellular Automata of the Two-Dimensional Plane, Complex Systems, 33(3), 2024, pp. 253-276.

Pattern replicating ETA rules: A275667 (rule 170).
A247640 is a bisection.
A246035 is the analog on the square cells.

A166976 Array of A002450 in the top row and higher-order differences in subsequent rows, read by antidiagonals.

Original entry on oeis.org

0, 1, 1, 3, 4, 5, 9, 12, 16, 21, 27, 36, 48, 64, 85, 81, 108, 144, 192, 256, 341, 243, 324, 432, 576, 768, 1024, 1365, 729, 972, 1296, 1728, 2304, 3072, 4096, 5461, 2187, 2916, 3888, 5184, 6912, 9216, 12288, 16384, 21845, 6561

Offset: 0

Paul Curtz, Oct 26 2009

			The array starts:
0,   1,   5,  21,  85, 341,1365,5461,21845,87381,349525,    A002450
1,   4,  16,  64, 256,1024,4096,16384,65536,262144,1048576, A000302
3,  12,  48, 192, 768,3072,12288,49152,196608,786432,       A002001, A164346, A110594
9,  36, 144, 576,2304,9216,36864,147456                     A002063, A055841

A002450 := proc(n) (4^n-1)/3 ; end proc:
A166976 := proc(n,k) option remember; if n = 0 then A002450(k) else procname(n-1,k+1)-procname(n-1,k) ; end if; end proc: # R. J. Mathar, Jul 02 2011

T(0,k) = A002450(k). T(n,k) = T(n-1,k+1) - T(n-1,k), n > 0.

cp's OEIS Frontend

A164346 a(n) = 3 * 4^n.

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A110591 Number of digits in base-4 representation of n.

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A092898 Expansion of (1 - 4*x + 4*x^2 - 4*x^3)/(1 - 4*x).

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A110595 a(1)=5. For n > 1, a(n) = 4*5^(n-1) = A005054(n).

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A383369 Population of elementary triangular automaton rule 90 at generation n, starting from a lone 1 cell at generation 0.

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Crossrefs

A166976 Array of A002450 in the top row and higher-order differences in subsequent rows, read by antidiagonals.

Views

Author

Keywords

Examples

Programs

Maple

Formula

A092898 Expansion of (1 - 4x + 4x^2 - 4x^3)/(1 - 4x).