A349276 -id:A349276 - cp's OEIS frontend

A255047 1 together with the positive terms of A000225.

1, 1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 4095, 8191, 16383, 32767, 65535, 131071, 262143, 524287, 1048575, 2097151, 4194303, 8388607, 16777215, 33554431, 67108863, 134217727, 268435455, 536870911, 1073741823, 2147483647, 4294967295

Offset: 0

Omar E. Pol, Feb 15 2015

Also, right border of A246674 arranged as an irregular triangle.
Essentially the same as A168604, A126646 and A000225.
Total number of lambda-parking functions induced by all partitions of n. a(0)=1: [], a(1)=1: [1], a(2)=3: [1], [2], [1,1], a(4)=7: [1], [2], [3], [1,1], [1,2], [2,1], [1,1,1]. - Alois P. Heinz, Dec 04 2015
Also, the 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 645", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. - Robert Price, Jul 19 2017
Also number of multiset partitions of {1,1} U [n] into exactly 2 nonempty parts.  a(2) = 3: 111|2, 11|12, 1|112. - Alois P. Heinz, Aug 18 2017
Also, the number of unlabeled connected P-series (equivalently, connected P-graphs) with n+1 elements. - Salah Uddin Mohammad, Nov 19 2021

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

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
R. Stanley, Parking Functions, 2011.
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 (3,-2).

Cf. A000225, A011782, A028310, A246674, A253909, A265007, A265202, A349276.
Row n=1 of A263159.
Column k=2 of A291117.
Cf. A078485.

[1] cat [2^n -1: n in [1..40]]; // G. C. Greubel, Feb 07 2021

CoefficientList[Series[(1 -2*x +2*x^2)/((1-x)*(1-2*x)), {x, 0, 33}], x] (* or *) LinearRecurrence[{3, -2}, {1,1,3}, 40] (* Vincenzo Librandi, Jul 20 2017 *)
Table[2^n -1 +Boole[n==0], {n, 0, 40}] (* G. C. Greubel, Feb 07 2021 *)

def A255047(n): return -1^(-1<Chai Wah Wu, Dec 21 2022

[1]+[2^n -1 for n in (1..40)] # G. C. Greubel, Feb 07 2021

From Alois P. Heinz, Feb 19 2015: (Start)
O.g.f.: (1 -2*x +2*x^2)/((1-x)*(1-2*x)).
E.g.f.: exp(2*x) - exp(x) + 1. (End)
a(n) = A078485(n+1) for n > 2. - Georg Fischer, Oct 22 2018

A349488 Number of unlabeled disconnected P-series with n elements.

Original entry on oeis.org

0, 1, 2, 6, 16, 45, 115, 296, 733, 1801, 4338, 10380, 24531, 57622, 134317, 311465, 718297, 1649579, 3772448, 8597284, 19527774, 44225665, 99885035, 225032910, 505797776, 1134419571, 2539173978, 5672736196, 12650878942, 28165845957, 62609097765, 138963709623

Offset: 1

Salah Uddin Mohammad, Nov 19 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..3217

Cf. A255047, A349276.

b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(d*
      max(1, 2^(d-1)-1), d=numtheory[divisors](j)), j=1..n)/n)
    end:
a:= n-> b(n)-max(1, 2^(n-1)-1):
seq(a(n), n=1..35);  # Alois P. Heinz, Jan 05 2022

b[n_] := b[n] = If[n == 0, 1, Sum[b[n - j]*Sum[d*
     Max[1, 2^(d-1) - 1], {d, Divisors[j]}], {j, 1, n}]/n];
a[n_] := b[n] - Max[1, 2^(n-1)-1];
Table[a[n], {n, 1, 35}] (* Jean-François Alcover, Mar 11 2022, Alois P. Heinz *)

a(n) = A349276(n) - A255047(n-1).

A350635 Triangle read by rows: T(n,k) is the number of n-element unlabeled P-series with k connected components.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 7, 4, 1, 1, 15, 10, 4, 1, 1, 31, 28, 11, 4, 1, 1, 63, 67, 31, 11, 4, 1, 1, 127, 167, 80, 32, 11, 4, 1, 1, 255, 388, 213, 83, 32, 11, 4, 1, 1, 511, 908, 534, 226, 84, 32, 11, 4, 1, 1, 1023, 2053, 1343, 580, 229, 84, 32, 11, 4, 1, 1

Offset: 1

Salah Uddin Mohammad, Jan 09 2022

			Triangle begins:
    1;
    1,   1;
    3,   1,  1;
    7,   4,  1,  1;
   15,  10,  4,  1,  1;
   31,  28, 11,  4,  1, 1;
   63,  67, 31, 11,  4, 1, 1;
  127, 167, 80, 32, 11, 4, 1, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Row sums give A349276.
Column 1 is A255047(n-1).
Cf.  A263864 (all posets), A349488 (disconnected).

B(x) = x*(1 - 2*x + 2*x^2)/((1 - x)*(1 - 2*x))
T(n)=[Vecrev(p/y) | p<-Vec(-1 + exp(sum(k=1, n, y^k*B(x^k)/k + O(x*x^n))))]
{ my(A=T(8)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 13 2022

G.f.: -1 + exp(Sum_{k>=1} y^k*B(x^k)/k) where B(x) = x*(1 - 2*x + 2*x^2)/((1 - x)*(1 - 2*x)). - Andrew Howroyd, Jan 13 2022

cp's OEIS Frontend

A255047 1 together with the positive terms of A000225.

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A349488 Number of unlabeled disconnected P-series with n elements.

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A350635 Triangle read by rows: T(n,k) is the number of n-element unlabeled P-series with k connected components.

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