OFFSET
0,3
COMMENTS
Also, right border of A246674 arranged as an irregular triangle.
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 linear recurrences with constant coefficients, signature (3,-2).
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
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 *)
PROG
(Sage) [1]+[2^n -1 for n in (1..40)] # G. C. Greubel, Feb 07 2021
(Magma) [1] cat [2^n -1: n in [1..40]]; // G. C. Greubel, Feb 07 2021
(Python)
def A255047(n): return -1^(-1<<n) if n else 1 # Chai Wah Wu, Dec 21 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Omar E. Pol, Feb 15 2015
STATUS
approved