A060576 -id:A060576 - cp's OEIS frontend

A267940 Binary representation of the n-th iteration of the "Rule 253" elementary cellular automaton starting with a single ON (black) cell.

1, 11, 11111, 1111111, 111111111, 11111111111, 1111111111111, 111111111111111, 11111111111111111, 1111111111111111111, 111111111111111111111, 11111111111111111111111, 1111111111111111111111111, 111111111111111111111111111, 11111111111111111111111111111

Offset: 0

Robert Price, Jan 22 2016

With the exception of a(1) the same as A267937, A267889, A267887 and A100706. - R. J. Mathar, Jan 24 2016

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

Robert Price, Table of n, a(n) for n = 0..1000
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. A060576.

rule=253; 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]]],{k,1,rows}]   (* Binary Representation of Rows *)

Conjectures from Colin Barker, Jan 23 2016 and Apr 16 2019: (Start)
a(n) = 101*a(n-1)-100*a(n-2) for n>3.
G.f.: (1-90*x+10100*x^2-10000*x^3) / ((1-x)*(1-100*x)).
(End)

A267941 Decimal representation of the n-th iteration of the "Rule 253" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 3, 31, 127, 511, 2047, 8191, 32767, 131071, 524287, 2097151, 8388607, 33554431, 134217727, 536870911, 2147483647, 8589934591, 34359738367, 137438953471, 549755813887, 2199023255551, 8796093022207, 35184372088831, 140737488355327, 562949953421311

Offset: 0

Robert Price, Jan 22 2016

With the exception of a(1) the same as A267938, A267890, A267888 and A083420. - R. J. Mathar, Jan 24 2016

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

Robert Price, Table of n, a(n) for n = 0..1000
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. A060576.

rule=253; 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 23 2016 and Apr 16 2019: (Start)
a(n) = 5*a(n-1)-4*a(n-2) for n>3.
G.f.: (1-2*x+20*x^2-16*x^3) / ((1-x)*(1-4*x)).
(End)
Empirical a(n) = 2^(2*n+1) - 1 for n>1. - Colin Barker, Nov 26 2016

A274922 a(n) = (-1)^n * n if n>0, a(0) = 1.

Original entry on oeis.org

1, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 18, -19, 20, -21, 22, -23, 24, -25, 26, -27, 28, -29, 30, -31, 32, -33, 34, -35, 36, -37, 38, -39, 40, -41, 42, -43, 44, -45, 46, -47, 48, -49, 50, -51, 52, -53, 54, -55, 56, -57, 58, -59

Offset: 0

Michael Somos, Dec 28 2016

This is a divisibility sequence.

			G.f. = 1 - x + 2*x^2 - 3*x^3 + 4*x^4 - 5*x^5 + 6*x^6 - 7*x^7 + 8*x^8 + ...

Index entries for linear recurrences with constant coefficients, signature (-2,-1).

Cf. A004277, A005408, A028310, A060576, A106510.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1 + x+x^2)/(1+2*x+x^2))); // G. C. Greubel, Jul 29 2018

a[ n_] := If[ n < 1, Boole[n == 0], (-1)^n n];
a[ n_] := SeriesCoefficient[ (1 + x + x^2) / (1 + 2*x + x^2), {x, 0, n}];
LinearRecurrence[{-2,-1},{1,-1,2},60] (* Harvey P. Dale, Mar 30 2019 *)

PARI
```
{a(n) = if( n<1, n==0, (-1)^n * n)};
```

{a(n) = if( n<0, 0, polcoeff( (1 + x + x^2) / (1 + 2*x + x^2) + x * O(x^n), n))};

Euler transform of length 3 sequence [-1, 2, -1].
a(n) = -b(n) where b() is multiplicative with b(2^e) = -(2^e) if e>0, b(p^e) = p^e otherwise.
E.g.f.: 1 - x * exp(-x).
G.f.: (1 + x + x^2) / (1 + 2*x + x^2).
G.f.: (1 - x) * (1 - x^3) / (1 - x^2)^2.
G.f.: 1 / (1 + x / (1 + x / (1 - x / (1 + x)))).
G.f.: 1 - x / (1 + x)^2 = 1 - x /(1 - x)^2 + 4*x^2 / (1 - x^2)^2.
a(n) = (-1)^n * A028310(n). a(2*n) = A004277(n). a(2*n + 1) = - A005408(n).
Convolution inverse of A106510.
A060576(n) = Sum_{k=0..n} binomial(n, k) * a(k).
A028310(n) = Sum_{k=0..n} binomial(n+1, k+1) * a(k).
a(n) = A038608(n), n>0. - R. J. Mathar, May 25 2020

A305559 [0, -1, -1] together with A000290.

Original entry on oeis.org

0, -1, -1, 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 2401, 2500

Offset: 0

Paul Curtz, Jun 21 2018

Squares leading to an autosequence of the first kind.
The third sequence of the array
A060576(n+1)=         0,  1, 1,  1,  1,   1,   1,   1,   1,  1, ...
A289207(n)=      0,   0,  0, 1,  2,  3,   4,   5,   6,   7, ...
a(n)=       0,  -1,  -1,  0, 1,  4,  9,  16,  25,  36, ...
      0,   10,  10,   5,  0, 1,  8, 27,  64, 125, ...
0, -113, -113, -68, -23,  0, 1, 16, 81, 256, ... .
The first full vertical is (-1)^n*A033312(n).
From 0, the first two nonzero antidiagonals are 0, -1, 10, -113, 1526, ... = (-1)^n* A176824(n+1).
See OEIS Wiki, Autosequence.
a(n) difference table:
0, -1, -1, 0, 1, 4, 9, 16, 25, ...
-1, 0,  1, 1, 3, 5, 7,  9, 11, ...
1,  1,  0, 2, 2, 2, 2,  2,  2, ...
0, -1,  2, 0, 0, 0, 0,  0,  0, ...

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A003992, A033312, A060576, A176824, A289207, A000290, A113679.

Join[{0,-1,-1},Range[0,100]^2] (* Paolo Xausa, Nov 13 2023 *)

From Stefano Spezia, May 28 2025: (Start)
G.f.: x*(1 - 2*x + x^3 - 2*x^4)/(1 - x)^3.
E.g.f.: 9 + 5*x + x^2 - exp(x)*(9 - 5*x + x^2). (End)

A060578 Number of homeomorphically irreducible general graphs on 3 labeled node and with n edges.

Original entry on oeis.org

1, 3, 9, 21, 60, 135, 282, 537, 945, 1561, 2451, 3693, 5378, 7611, 10512, 14217, 18879, 24669, 31777, 40413, 50808, 63215, 77910, 95193, 115389, 138849, 165951, 197101, 232734, 273315, 319340, 371337, 429867, 495525, 568941, 650781, 741748

Offset: 0

Vladeta Jovovic, Apr 03 2001

A homeomorphically irreducible general graph is a graph with multiple edges and loops and without nodes of degree 2.

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983.

Cf. A003514, A060516, A060533-A060537, A060576-A060581.

CoefficientList[Series[-(8x^9-36x^8+66x^7-70x^6+51x^5-24x^4+8x^3-6x^2+3x-1)/(x-1)^6,{x,0,40}],x] (* Harvey P. Dale, Jul 22 2018 *)

G.f.: - (8*x^9 - 36*x^8 + 66*x^7 - 70*x^6 + 51*x^5 - 24*x^4 + 8*x^3 - 6*x^2 + 3*x - 1)/(x - 1)^6. E.g.f. for homeomorphically irreducible general graphs with n nodes and k edges is (1 + x*y)^( - 1/2)*exp( - x*y/2 + x^2*y^2/4)*Sum_{k >= 0} 1/(1 - x)^binomial(k + 1, 2)*exp( - x^2*y*k^2/(2*(1 + x*y)) - x^2*y*k/2)*y^k/k!.

A060579 Number of homeomorphically irreducible general graphs on 4 labeled nodes and with n edges.

Original entry on oeis.org

1, 6, 19, 68, 242, 704, 1981, 5140, 12364, 27614, 57598, 113108, 210812, 375606, 643646, 1066196, 1714445, 2685464, 4109493, 6158768, 9058119, 13097592, 18647371, 26175300, 36267330, 49651242, 67224024, 90083308, 119563302

Offset: 0

Vladeta Jovovic, Apr 03 2001

A homeomorphically irreducible general graph is a graph with multiple edges and loops and without nodes of degree 2.

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983.

Cf. A003514, A060516, A060533-A060537, A060576-A060581.

G.f.: (4*x^15 + 5*x^14 - 194*x^13 + 881*x^12 - 2058*x^11 + 3096*x^10 - 3330*x^9 + 2628*x^8 - 1398*x^7 + 359*x^6 + 72*x^5 - 93*x^4 + 28*x^3 + 4*x^2 - 4*x + 1)/(x - 1)^10. E.g.f. for homeomorphically irreducible general graphs with n nodes and k edges is (1 + x*y)^( - 1/2)*exp( - x*y/2 + x^2*y^2/4)*Sum_{k >= 0} 1/(1 - x)^binomial(k + 1, 2)*exp( - x^2*y*k^2/(2*(1 + x*y)) - x^2*y*k/2)*y^k/k!.

A060580 Number of homeomorphically irreducible general graphs on 5 labeled nodes and with n edges.

Original entry on oeis.org

1, 10, 40, 185, 765, 2845, 10220, 33885, 105185, 305465, 830811, 2119875, 5091525, 11565505, 24977315, 51552005, 102175360, 195301015, 361365695, 649360880, 1136438375, 1941722170, 3245874555, 5318438260, 8555568895, 13531506921

Offset: 0

Vladeta Jovovic, Apr 03 2001

A homeomorphically irreducible general graph is a graph with multiple edges and loops and without nodes of degree 2.

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983.

Cf. A003514, A060516, A060533-A060537, A060576-A060581.

G.f.: - (5*x^22 - 20*x^21 + 23*x^20 - 815*x^19 + 8110*x^18 - 37255*x^17 + 104890*x^16 - 204469*x^15 + 296720*x^14 - 337455*x^13 + 310150*x^12 - 229885*x^11 + 131054*x^10 - 50485*x^9 + 6490*x^8 + 7255*x^7 - 6730*x^6 + 3242*x^5 - 995*x^4 + 180*x^3 - 5*x^2 - 5*x + 1)/(x - 1)^15. E.g.f. for homeomorphically irreducible general graphs with n nodes and k edges is (1 + x*y)^( - 1/2)*exp( - x*y/2 + x^2*y^2/4)*Sum_{k >= 0} 1/(1 - x)^binomial(k + 1, 2)*exp( - x^2*y*k^2/(2*(1 + x*y)) - x^2*y*k/2)*y^k/k!.

A167371 Triangle, read by rows, given by [0,1,-1,0,0,0,0,0,0,0,0,...] DELTA [1,0,-1,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

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

Offset: 0

Philippe Deléham, Nov 02 2009

Diagonal sums: A060576.

A167374*A154325 formatted as lower triangular matrix. - Philippe Deléham, Nov 19 2009

			Triangle begins:
  1;
  0, 1;
  0, 1, 1;
  0, 0, 1, 1;
  0, 0, 0, 1, 1;
  0, 0, 0, 0, 1, 1; ...

Cf. A097806, A103451.

Sum_{k=0..n} T(n,k)*x^k = A000007(n), A046698(n+1), A111286(n+1), A027327(n) for x= 0, 1, 2, 3 respectively.
G.f.: (1+x^2*y)/(1-x*y). - Philippe Deléham, Nov 09 2013
T(n,k) = T(n-1,k-1) for n > 2, T(0,0) = T(1,1) = T(2,1) = T(2,2) = 1, T(1,0) = T(2,0) = 0, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Nov 09 2013

A273153 a(n) = Numerator of (0 followed by 1's) - n/2^n.

Original entry on oeis.org

0, 1, 1, 5, 3, 27, 29, 121, 31, 503, 507, 2037, 1021, 8179, 8185, 32753, 4095, 131055, 131063, 524269, 262139, 2097131, 2097141, 8388585, 2097149, 33554407, 33554419, 134217701, 67108857, 536870883, 536870897, 2147483617, 134217727, 8589934559, 8589934575, 34359738333

Offset: 0

Paul Curtz, May 16 2016

A060576(n+1) = 0, 1, 1, 1, 1, 1, 1, ... - (0(n) = Oresme(n) = 0, 1/2, 1/2, 3/8, 1/4, 5/32, 3/32, ...). Both sequences are autosequences of the first kind. f(n) = 0, 1/2, 1/2, 5/8, 3/4, 27/32, 29/32, 121/128, ... is an autosequence of the first kind. Without one 1/2, f(n) is an increasing sequence.

The numerators (1 followed by A075101(n)) are the same as in n/2^n.

			Array of differences of fractions (characteristic aspect of an autosequence of the first kind):
0,     1/2,   1/2,   5/8,   3/4, ...
1/2,     0,   1/8,   1/8,  3/32, ...
-1/2,  1/8,     0, -1/32, -1/32, ...
5/8,  -1/8, -1/32,     0, 1/128, ...
-3/4, 3/32,  1/32, 1/128,     0, ...
...

OEIS Wiki, Autosequence

Cf. A060576, A054977, A075101, A209308.

{0}~Join~Array[Numerator@ Abs[1 - Binomial[0, # - 1] - #/2^#] &, 30] (* Michael De Vlieger, May 17 2016 *)

A368684 Number of partitions of n into 2 parts such that the smaller part divides both n and floor(n/2).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 4, 1, 3, 1, 4, 1, 2, 1, 6, 1, 2, 1, 4, 1, 4, 1, 5, 1, 2, 1, 6, 1, 2, 1, 6, 1, 4, 1, 4, 1, 2, 1, 8, 1, 3, 1, 4, 1, 4, 1, 6, 1, 2, 1, 8, 1, 2, 1, 6, 1, 4, 1, 4, 1, 4, 1, 9, 1, 2, 1, 4, 1, 4, 1, 8, 1, 2, 1, 8, 1, 2, 1, 6, 1, 6

Offset: 1

Wesley Ivan Hurt, Jan 03 2024

Essentially, A000005 interspersed with 1's [prepend 0].

Number of divisors of A057979(n+1) for n >= 2.

Index entries for sequences related to partitions.

Bisections: A060576, A000005.

Cf. A001620, A057979, A091954.

with(numtheory): 0, seq(2*tau(n) - tau(2*n) + (n mod 2), n=2..100); # Ridouane Oudra, Jan 18 2025

Join[{0}, Table[DivisorSigma[0, (n+2+(n-2)*(-1)^n)/4], {n, 2, 100}]]

a(n) = if(n == 1, 0, numdiv((n+2+(n-2)*(-1)^n)/4)); \\ Amiram Eldar, Jan 28 2025

a(n) = A000005(A057979(n+1)) for n >= 2.
a(2n-1) = A060576(n), a(2n) = A000005(n).
a(n) = d(floor((n+1)/2))^((n+1) mod 2), for n >= 2.
a(n) = d( (n+2+(n-2)*(-1)^n)/4 ) for n >= 2.
a(n) = Sum_{k=1..floor(n/2)} c(n/k) * c(floor(n/2)/k), where c(m) = 1 - ceiling(m) + floor(m).
a(n) = A000005(n) - A091954(n), for n > 1. - Ridouane Oudra, Jan 18 2025
Sum_{k=1..n} a(k) ~ (log(n/2) + 2*gamma)*n/2, where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 28 2025

cp's OEIS Frontend

A267940 Binary representation of the n-th iteration of the "Rule 253" elementary cellular automaton starting with a single ON (black) cell.

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A267941 Decimal representation of the n-th iteration of the "Rule 253" elementary cellular automaton starting with a single ON (black) cell.

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A274922 a(n) = (-1)^n * n if n>0, a(0) = 1.

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A305559 [0, -1, -1] together with A000290.

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A060578 Number of homeomorphically irreducible general graphs on 3 labeled node and with n edges.

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A060579 Number of homeomorphically irreducible general graphs on 4 labeled nodes and with n edges.

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A060580 Number of homeomorphically irreducible general graphs on 5 labeled nodes and with n edges.

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A167371 Triangle, read by rows, given by [0,1,-1,0,0,0,0,0,0,0,0,...] DELTA [1,0,-1,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

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