A024036 -id:A024036 - cp's OEIS frontend

A185172 Dimensions of primitive Lie algebras connected with the Mantaci-Reutenauer algebra MR^(3).

3, 6, 20, 60, 204, 670, 2340, 8160, 29120, 104754, 381300, 1397740, 5162220, 19172790, 71582716, 268431360, 1010580540, 3817733920, 14467258260, 54975528948, 209430785460, 799644629550, 3059510616420, 11728123327840, 45035996273664, 173215367702370

Offset: 1

N. J. A. Sloane, Jan 23 2012

Maybe the definition should say: "Number of generators of degree n ...". The paper is a little unclear.
From Petros Hadjicostas, Jun 18 2019: (Start)
An unmarked cyclic composition of n >= 1 is an equivalence class of ordered partitions of n such that two such ordered partitions are equivalent iff one can be obtained from the other by rotation.
Here, a(n) is the number of aperiodic unmarked cyclic compositions of n where up to three colors can be used.
It is also the CHK (circular, identity, unlabeled) transform of the sequence 3, 3, 3, ... See the link by Bowers about such transforms.
If c = (c(n): n >= 1) is the input sequence with g.f. C(x) = Sum_{n >= 1} c(n)*x^n, then the g.f. of the output sequence ((CHK c)d: d >= 1) is -Sum{d >= 1} (mu(d)/d) * log(1 - C(x^d)). Here, c(n) = 3 for all n >= 1, and thus, C(x) = 3*x/(1 - x). It follows that the g.f. of the output sequence is -Sum_{d >= 1} (mu(d)/d) * log(1 - 3*x^d/(1 - x^d)).
(End)

			From _Petros Hadjicostas_, Jun 18 2019: (Start)
Suppose we have three colors, say, A, B, C. Here, a(1) = 3 because we have the following aperiodic unmarked cyclic compositions of n = 1: 1_A, 1_B, 1_C.
We have a(2) = 6 because we have the following aperiodic unmarked cyclic compositions of n = 2: 2_A, 2_B, 2_C, 1_A + 1_B, 1_B + 1_C, 1_C + 1_A.
We have a(3) = 20 because we have the following aperiodic unmarked cyclic compositions of n = 3: 3_X, where X \in {A, B, C}; 1_X + 2_Y, where (X, Y) \in {(A, A), (A, B), (A, C), (B, A), (B, B), (B, C), (C, A), (C, B), (C, C)}; 1_A + 1_B + 1_C and 1_C + 1_B + 1_A; and 1_X + 1_Y + 1_Y, where (X, Y) \in {(A, B), (A, C), (B, A), (B, C), (C, A), (C, B)}.
(End)

G. C. Greubel, Table of n, a(n) for n = 1..1650
C. G. Bower, Transforms (2).
Jean-Christophe Novelli and Jean-Yves Thibon, Free quasi-symmetric functions and descent algebras for wreath products, and noncommutative multi-symmetric functions, arXiv:0806.3682 [math.CO], 2008. See Eqs. (94) and (95).
Jean-Christophe Novelli and Jean-Yves Thibon, Free quasi-symmetric functions and descent algebras for wreath products, and noncommutative multi-symmetric functions, Discrete Math. 310 (2010), no. 24, 3584-3606. See Eqs. (99) and (100).

Cf. A024036, A032253, A185171.

Essentially the same as A027377.

a[1] = 3; a[n_] := DivisorSum[n, MoebiusMu[#]*4^(n/#)&]/n; Array[a, 26] (* Jean-François Alcover, Dec 07 2015, adapted from PARI *)

a(l=3,n) = if (n==1, l, sumdiv(n, d, moebius(d)*(1+l)^(n/d))/n);  \\ Michel Marcus, Feb 09 2013

From Petros Hadjicostas, Jun 17 2019: (Start)
a(1) = 3 and a(n) = (1/n) * Sum_{d|n} mu(d) * 4^(n/d) for n > 1 (from Eq. (95) in Novelli and Thibon (2008) or Eq. (100) in Novelli and Thibon (2010)).
a(n) = (1/n) * Sum_{d|n} mu(d) * (4^(n/d) - 1) = (1/n) * Sum_{d|n} mu(d) *A024036(n/d) for n >= 1.
G.f.: -Sum_{d >= 1} (mu(d)/d) * log(1 - 3*x^d/(1 - x^d)) = -x - Sum_{d >= 1} (mu(d)/d) * log(1 - 4*x^d).
(End)

More terms from Michel Marcus, Feb 09 2013

Name edited by Petros Hadjicostas, Jun 17 2019

A198694 a(n) = 7*4^n-1.

Original entry on oeis.org

6, 27, 111, 447, 1791, 7167, 28671, 114687, 458751, 1835007, 7340031, 29360127, 117440511, 469762047, 1879048191, 7516192767, 30064771071, 120259084287, 481036337151, 1924145348607, 7696581394431, 30786325577727, 123145302310911, 492581209243647, 1970324836974591

Offset: 0

Vincenzo Librandi, Oct 29 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A024036, A097743, A114569, A156760.

Magma
```
[7*4^n-1: n in [0..30]]
```

7*4^Range[0,30]-1 (* or *) LinearRecurrence[{5,-4},{6,27},30] (* Harvey P. Dale, Nov 14 2018 *)

a(n) = 4*a(n-1)+3.
a(n) = 5*a(n-1)-4*a(n-2), n>1.
G.f.: ( 6-3*x ) / ( (4*x-1)*(x-1) ). - R. J. Mathar, Oct 30 2011
E.g.f.: exp(x)*(7*exp(3*x) - 1). - Stefano Spezia, Apr 17 2024

A248337 a(n) = 6^n - 4^n.

Original entry on oeis.org

0, 2, 20, 152, 1040, 6752, 42560, 263552, 1614080, 9815552, 59417600, 358602752, 2160005120, 12993585152, 78095728640, 469111242752, 2816814940160, 16909479575552, 101491237191680, 609084862103552, 3655058928435200, 21932552593866752, 131604111656222720, 789659854309425152, 4738099863344906240, 28429162130022858752

Offset: 0

Vincenzo Librandi, Oct 05 2014

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-24).

Cf. sequences of the form k^n - 4^n: -A000302 (k=0), -A024036 (k=1), -A020522 (k=2), -A005061 (k=3), A005060 (k=5), this sequence (k=6), A190542 (k=7), A059409 (k=8), A118004 (k=9), A248338 (k=10), A139742 (k=11), 8*A016159 (k=12).

Cf. A000079, A000302, A000400, A001047, A081199.

Magma
```
[6^n-4^n: n in [0..30]];
```

Table[6^n - 4^n, {n,0,30}]
CoefficientList[Series[(2 x)/((1-4 x)(1-6 x)), {x, 0, 30}], x]
LinearRecurrence[{10,-24},{0,2},30] (* Harvey P. Dale, Aug 18 2024 *)

vector(20,n,6^(n-1)-4^(n-1)) \\ Derek Orr, Oct 05 2014

A248337=BinaryRecurrenceSequence(10,-24,0,2)
[A248337(n) for n in range(31)] # G. C. Greubel, Nov 11 2024

G.f.: 2*x/((1-4*x)*(1-6*x)).
a(n) = 10*a(n-1) - 24*a(n-2).
a(n) = 2^n*(3^n-2^n) =  A000079(n) * A001047(n) = A000400(n) - A000302(n).
a(n) = 2*A081199(n). - Bruno Berselli, Oct 05 2014
E.g.f.: 2*exp(5*x)*sinh(x). - G. C. Greubel, Nov 11 2024

More terms added by G. C. Greubel, Nov 11 2024

A285475 Decimal representation of the diagonal from the origin to the corner of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 3", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 3, 4, 15, 16, 63, 64, 255, 256, 1023, 1024, 4095, 4096, 16383, 16384, 65535, 65536, 262143, 262144, 1048575, 1048576, 4194303, 4194304, 16777215, 16777216, 67108863, 67108864, 268435455, 268435456, 1073741823, 1073741824, 4294967295, 4294967296

Offset: 0

Robert Price, Apr 19 2017

Initialized with a single black (ON) cell at stage zero.

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Robert Price, Table of n, a(n) for n = 0..126
Robert Price, Diagrams of first 20 stages
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Wolfram Research, Wolfram Atlas of Simple Programs
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata
Index entries for linear recurrences with constant coefficients, signature (0,5,0,-4).

Cf. A285473, A285474, A080924.
Cf. A083420.
Cf. A000302 (even bisection), A024036 (odd bisection).

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 3; 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]], 10], {i, 1, stages - 1}]

From Colin Barker, Apr 19 2017: (Start)
G.f.: (1 + 3*x - x^2) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)).
a(n) = (-1 - (-2)^n + (-1)^n + 3*2^n)/2.
a(n) = 5*a(n-2) - 4*a(n-4) for n>3. (End)
a(2*n-1) + a(2*n) = A083420(n). - Paul Curtz, Dec 16 2024

A330940 a(n) is the least value whose binary representation can be obtained by interleaving (or shuffling) two copies of the binary representation of n.

Original entry on oeis.org

0, 3, 10, 15, 36, 43, 54, 63, 136, 147, 170, 175, 204, 219, 238, 255, 528, 547, 586, 591, 660, 683, 694, 703, 792, 819, 858, 879, 924, 955, 990, 1023, 2080, 2115, 2186, 2191, 2340, 2347, 2358, 2367, 2600, 2643, 2730, 2735, 2764, 2779, 2798, 2815, 3120, 3171

Offset: 0

Rémy Sigrist, Jan 04 2020

The binary representation of all positive terms are square binary words (see A191755).

			The first terms, alongside the binary representation of n and of a(n), are:
  n   a(n)  bin(n)  bin(a(n))
  --  ----  ------  ---------
   0     0       0          0
   1     3       1         11
   2    10      10       1010
   3    15      11       1111
   4    36     100     100100
   5    43     101     101011
   6    54     110     110110
   7    63     111     111111
   8   136    1000   10001000
   9   147    1001   10010011
  10   170    1010   10101010
  11   175    1011   10101111

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Rémy Sigrist, Logarithmic scatterplot of the first difference of the first 2^13 terms
Rémy Sigrist, PARI program for A330940
Index entries for sequences related to binary expansion of n

See A330941 for the maximum variant.

Cf. A007582, A024036, A191755, A193020.

PARI
```
See Links section.
```

a(2^k) = 2^k*(1+2^(k+1)) = A007582(k+1) for any k >= 0.
a(2^k-1) = 4^k-1 = A024036(k) for any k >= 0.
a(n) <= A330941(n).

A175824 Maximum unsigned integer that can be stored in n bytes.

Original entry on oeis.org

0, 255, 65535, 16777215, 4294967295, 1099511627775, 281474976710655, 72057594037927935, 18446744073709551615, 4722366482869645213695, 1208925819614629174706175, 309485009821345068724781055

Offset: 0

Grant Garcia, Sep 13 2010

All a(n) for n > 0 are Mersenne numbers. None are Mersenne primes.

			a(0) = 2^0 - 1 = 1 - 1 = 0
a(1) = 2^8 - 1 = 256 - 1 = 255
a(2) = 2^16 - 1 = 65536 - 1 = 65535
a(3) = 2^24 - 1 = 16777216 - 1 = 16777215

Wikipedia, Byte
Wikipedia, Mersenne prime
Index entries for linear recurrences with constant coefficients, signature (257, -256).

Cf. A000225. Equals A133752 minus one.

Mathematica
```
Table[2^(8n) - 1, {n, 0, 11}]
```

print([2**(8 * i) - 1 for i in range(12)])

a(n) = 2^(8*n) - 1.

a(n) = (A001025(n) - 1) * (A024036(n)^2 + A004171(n)); this relation is (x^(8*n)-1) = (x^(4*n)-1)*((x^(2*n)-1)^2 + 2*x^(2*n)) for x=2. [Reinhard Zumkeller, Jun 23 2011]

A267596 Decimal representation of the n-th iteration of the "Rule 173" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 2, 15, 63, 255, 1023, 4095, 16383, 65535, 262143, 1048575, 4194303, 16777215, 67108863, 268435455, 1073741823, 4294967295, 17179869183, 68719476735, 274877906943, 1099511627775, 4398046511103, 17592186044415, 70368744177663, 281474976710655

Offset: 0

Robert Price, Jan 18 2016

Robert Price, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Cf. A267594, A267595.

rule=173; rows=20; ca=CellularAutomaton[rule,{{1},0},rows-1,{All,All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]],{rows-k+1,rows+k-1}],{k,1,rows}]; (* Truncated list of each row *) Table[FromDigits[catri[[k]],2],{k,1,rows}]   (* Decimal Representation of Rows *)

Conjectures from Colin Barker, Jan 18 2016 and Apr 20 2019: (Start)
a(n) = 5*a(n-1)-4*a(n-2) for n>3.
G.f.: (1-3*x+9*x^2-4*x^3) / ((1-x)*(1-4*x)).
(End)
Empirical a(n) = 4^n - 1 = A103454(n) = A024036(n) for n>1. - Colin Barker, Nov 25 2016 and Apr 20 2019

A323203 "Primitive" numbers k such that k divides 4^k - 1.

Original entry on oeis.org

1, 3, 21, 147, 171, 657, 903, 1029, 1197, 2667, 3249, 4599, 6321, 7077, 7203, 8379, 12483, 13203, 18669, 22743, 32193, 38829, 44247, 47961, 49539, 50421, 51471, 58653, 61731, 71631, 87381, 92421, 97641, 113799, 114681, 118341, 130683, 152019, 159201, 197757

Offset: 1

Bernard Schott, Jan 07 2019

In the comments of A014945, Charles R. Greathouse writes "this sequence is closed under multiplication". So, here, the terms are only the "primitive" integers which satisfy the definition and are not the product of two or more previous numbers of the sequence. This sequence is a subsequence of A014945.
Also numbers k in A014945 such that no divisors d > 1 of k exist where d and k/d are in A014945. - David A. Corneth, Jan 11 2019
Following an observation of David A. Corneth, yes, a(n) is divisible by 3 for n > 1, there is a proof by Robert Israel in A014945. - Bernard Schott, Jan 25 2019

			3 is a term because 3 * 21 = 4^3 - 1.
63 divides 4^63 - 1, but 63 is not a term because 63 = 3 * 21 with 3 which divides 4^3 - 1, and 21 which divides 4^21 - 1.

Amiram Eldar, Table of n, a(n) for n = 1..3200

Cf. A014945, A024036.

filter:= proc(n) local d;
  if 4 &^ n - 1 mod n <> 0 then return false fi;
  for d in select(t -> t > 1 and t^2 <= n, numtheory:-divisors(n)) do
    if 4 &^ d - 1 mod d = 0 and 4 &^ (n/d) - 1 mod (n/d) = 0 then return false fi;
  od;
true
end proc:
select(filter, [$1..200000]); # Robert Israel, Jan 24 2019

is(n) = my(d=divisors(n)); if(Mod(4,n)^n != 1, return(0)); for(i = 2, (#d - 1) >> 1 + 1, if(Mod(4,d[i]) ^ d[i] == 1 && Mod(4, n/d[i]) ^ (n/d[i])==1, return(0))); 1
first(n) = n = max(n, 2); my(res = vector(n), t=1); res[1] = 1;forstep(i = 3, oo, 3, if(is(i), t++; res[t] = i; if(t==n, return(res)))) \\ David A. Corneth, Jan 11 2019

More terms (using b-file for A014945) from Jon E. Schoenfield, Jan 11 2019

Terms verified by Jon E. Schoenfield and David A. Corneth, Jan 12 2019

A060242 a(n) = (2^n - 1)*(4^n - 1).

Original entry on oeis.org

0, 3, 45, 441, 3825, 31713, 257985, 2080641, 16711425, 133955073, 1072692225, 8585738241, 68702695425, 549688696833, 4397778059265, 35183298314241, 281470681677825, 2251782633684993, 18014329789743105, 144114913197424641, 1152920405094170625

Offset: 0

N. J. A. Sloane, Mar 22 2001

Harry J. Smith, Table of n, a(n) for n = 0..100
Index entries for linear recurrences with constant coefficients, signature (15,-70,120,-64).

Cf. A000225, A024036.

[(2^n-1)^2*(2^n+1): n in [0..40]]; // G. C. Greubel, Nov 05 2024

f:= gfun:-rectoproc({a(n) - 15*a(n-1) + 70*a(n-2) - 120*a(n-3) + 64*a(n-4) = 0, a(0) = 0, a(1)=3, a(2)=45, a(3)=441}, a(n), remember): map(f, [$0..20]); # Georg Fischer, Feb 19 2021

Table[(2^n-1)(4^n-1),{n,0,20}] (* Harvey P. Dale, May 13 2019 *)

a(n) = (2^n - 1)*(4^n - 1); \\ Harry J. Smith, Jul 04 2009

def A060242(n): return (2^n-1)*(4^n-1)
[A060242(n) for n in range(41)] # G. C. Greubel, Nov 05 2024

G.f.: 3*x*(1-8*x^2)/((1-x)*(1-2*x)*(1-4*x)*(1-8*x)). - Alois P. Heinz, Feb 19 2021

E.g.f.: exp(x) - exp(2*x) - exp(4*x) + exp(8*x). - G. C. Greubel, Nov 05 2024

A226379 a(5n) = 2n(2n+1), a(5n+1) = (2n-3)(2n+5), a(5n+2) = (2n-1)(2n+3), a(5n+3) = (2n+2)(2n+1), a(5n+4) = (2n+1)(2*n+3).

Original entry on oeis.org

0, -15, -3, 2, 3, 6, -7, 5, 12, 15, 20, 9, 21, 30, 35, 42, 33, 45, 56, 63, 72, 65, 77, 90, 99, 110, 105, 117, 132, 143, 156, 153, 165, 182, 195, 210, 209, 221, 240, 255, 272, 273, 285, 306, 323, 342, 345, 357, 380, 399, 420, 425, 437

Offset: 0

Paul Curtz, Jun 05 2013

The sequence is the fifth row of the following array:
0,   6,  20, 42, 72, 110, 156, 210, 272, ...   A002943
0,   3,   6, 15, 20,  35,  42,  63,  72, ...   bisections A002943, A000466
0,   2,   3,  6, 12,  15,  20,  30,  35, ...   A226023 (trisections A002943, A000466, A002439)
0,  -3,   2,  3,  6,   5,  12,  15,  20, ...   A214297 (quadrisections A078371)
0, -15,  -3,  2,  3,   6,  -7,   5,  12, ...   a(n)
0, -63, -15, -3,  2,   3,   6, -55,  -7, ...
The principle of construction is that (i) the lower left triangular portion has constant values down the diagonals (6, 3, 2, -3, -15, ...), defined from row 4 on by the negated values of A024036. (ii) The extension along the rows is defined by maintaining bisections, trisections, quadrisections etc of the form (2*n+x)*(2*n+y) with some constants x and y. In the fifth line this needs the quintisections shown in the NAME.
Each row in the array has the subsequences of the previous row plus another subsequence of the format (2*n+1)*(2*n+y) shuffled in; the first A002943, the second also A000466, the third also A002439, the fourth also A078371, and the fifth (2*n+3)*(2*n-5).
Only the first three rows are monotonically increasing everywhere.
a(n) is divisible by A226203(n).
Numerators of: 0, -15/4, -3/4, 2/9, 3/16, 6/25, -7/36, 5/36, 12/49, 15/64, 20/81, ... = a(n)/A226096(n). A permutation of A225948(n+1)/A226008(n+1).
Is the sequence increasing monotonically from 221 on?

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,2,-2,0,0,0,-1,1).

Cf. A000466, A002439, A002939, A002943, A024036, A078371, A145923.

Cf. A214297, A225948, A226008, A226023, A226096, A226203.

R:=PowerSeriesRing(Integers(), 60); [0] cat Coefficients(R!( -x*(15-12*x-5*x^2-x^3-3*x^4-17*x^5+12*x^6+3*x^7-x^8+x^9)/((1-x^5)^2*(1-x)) )); // G. C. Greubel, Mar 23 2024

CoefficientList[Series[x*(15 - 12*x - 5*x^2 - x^3 - 3*x^4 - 17*x^5 + 12*x^6 + 3*x^7 - x^8 + x^9)/((x^4 + x^3 + x^2 + x + 1)^2*(x - 1)^3), {x, 0, 80}], x] (* Wesley Ivan Hurt, Oct 03 2017 *)

def A226379_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( -x*(15-12*x-5*x^2-x^3-3*x^4-17*x^5+12*x^6+3*x^7-x^8+x^9)/((1-x^5)^2*(1-x)) ).list()
A226379_list(50) # G. C. Greubel, Mar 23 2024

4*a(n) = A226096(n) - period 5: repeat [1, 64, 16, 1, 4].
G.f.: x*(15-12*x-5*x^2-x^3-3*x^4-17*x^5+12*x^6+3*x^7-x^8+x^9) / ( (x^4+x^3+x^2+x+1)^2 *(x-1)^3 ). - R. J. Mathar, Jun 13 2013
a(n) = a(n-1)+2*a(n-5)-2*a(n-6)-a(n-10)+a(n-11) for n > 10. - Wesley Ivan Hurt, Oct 03 2017

cp's OEIS Frontend

A185172 Dimensions of primitive Lie algebras connected with the Mantaci-Reutenauer algebra MR^(3).

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A198694 a(n) = 7*4^n-1.

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Magma

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A248337 a(n) = 6^n - 4^n.

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Magma

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A285475 Decimal representation of the diagonal from the origin to the corner of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 3", based on the 5-celled von Neumann neighborhood.

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A330940 a(n) is the least value whose binary representation can be obtained by interleaving (or shuffling) two copies of the binary representation of n.

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PARI

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A175824 Maximum unsigned integer that can be stored in n bytes.

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Mathematica

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A267596 Decimal representation of the n-th iteration of the "Rule 173" elementary cellular automaton starting with a single ON (black) cell.

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Formula

A226379 a(5n) = 2n(2n+1), a(5n+1) = (2n-3)(2n+5), a(5n+2) = (2n-1)(2n+3), a(5n+3) = (2n+2)(2n+1), a(5n+4) = (2n+1)(2*n+3).