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%I #25 Jul 03 2023 09:00:17
%S -1,1,2,7,14,31,62,127,254,511,1022,2047,4094,8191,16382,32767,65534,
%T 131071,262142,524287,1048574,2097151,4194302,8388607,16777214,
%U 33554431,67108862,134217727,268435454,536870911
%N a(2*n) = 2*(2*4^(n-1)-1) and a(2*n-1) = 2*4^(n-1)-1.
%C The inverse binomial transform is 1, 1, 4, -2, 10, -14, 34, -62 which leads to (-1)^(n+1)*A135440(n).
%C For n > 0: A266161(a(n)) = n and A266161(m) < n for m < a(n). - _Reinhard Zumkeller_, Dec 22 2015
%C Also, the decimal representation of the diagonal from the corner to the origin of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 673", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. - _Robert Price_, Jul 23 2017
%D S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H Reinhard Zumkeller, <a href="/A140253/b140253.txt">Table of n, a(n) for n = 0..1000</a>
%H N. J. A. Sloane, <a href="http://arxiv.org/abs/1503.01168">On the Number of ON Cells in Cellular Automata</a>, arXiv:1503.01168 [math.CO], 2015
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H Wolfram Research, <a href="http://atlas.wolfram.com/">Wolfram Atlas of Simple Programs</a>
%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2, 1, -2).
%F a(2*n) = 2*A083420(n-1) and a(2*n+1) = A083420(n)
%F a(n+1) - a(n) = A014551(n); Jacobsthal-Lucas numbers.
%F a(2*n) + a(2*n+1) = 9*A002450(n)
%F a(n+1) - 2*a(n) = A010674(n+1); repeat 3, 0.
%F a(n) + A000034(n+1) = A000079(n); powers of 2.
%F a(n)= a(n-1) + 2*a(n-2) + 3. - _Gary Detlefs_, Jun 22 2010
%F a(n+1) = A000069(2^n); odious numbers. - _Johannes W. Meijer_, Jun 24 2011
%F a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n>2, a(0) = -1, a(1) = 1, a(2) = 2. - _Philippe Deléham_, Feb 25 2012
%F G.f.: (x^2+3*x-1)/((1-2*x)*(1-x)*(1+x)). - _Philippe Deléham_, Feb 25 2012
%p A140253:=proc(n): if type(n, odd) then 2*4^(((n+1)/2)-1)-1 else 2*(2*4^((n/2)-1)-1) fi: end: seq(A140253(n),n=0..29); # _Johannes W. Meijer_, Jun 24 2011
%t Table[(2^(n+1) - 3 + (-1)^(n+1))/2, {n, 0, 30}] (* _Jean-François Alcover_, Jun 05 2017 *)
%o (Haskell)
%o import Data.List (transpose)
%o a140253 n = a140253_list !! n
%o a140253_list = -1 : concat
%o (transpose [a083420_list, map (* 2) a083420_list])
%o -- _Reinhard Zumkeller_, Dec 22 2015
%Y Cf. A000034, A000069, A000079, A002450, A010674, A014551, A083420, A266161.
%K sign,easy
%O 0,3
%A _Paul Curtz_, Jun 23 2008
%E Edited, corrected and information added by _Johannes W. Meijer_, Jun 24 2011
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