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

Author

Omar E. Pol, Feb 15 2015

Keywords

• nonn
• easy

Comments

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

References

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

Links

• 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).

Crossrefs

Cf. A000225, A011782, A028310, A246674, A253909, A265007, A265202, A349276.

Row n=1 of A263159.

Column k=2 of A291117.

Cf. A078485.

Programs

Magma

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

Mathematica

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 *)

Python

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

Sage

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

Formula

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