A072328 - OEIS

%I #44 Feb 17 2024 11:13:16

%S 3,0,2,6,2,10,14,14,34,42,62,110,146,234,366,526,834,1258,1886,2926,

%T 4402,6698,10254,15502,23650,36010,54654,83310,126674,192618,293294,

%U 445966,678530,1032554,1570462,2389614,3635570,5530538,8414798,12801678,19475874

%N a(n+1) = 2*a(n-2) + a(n-1), with a(0) = 3, a(1) = 0, and a(2) = 2.

%C With the term indexed as shown, has property that p prime => p divides a(p).

%C a(n) = x^n + y^n + z^n with x, y, z the three roots of x^3 - x - 2. - _James R. Buddenhagen_, Nov 05 2013

%H Matthew Macauley , Jon McCammond, and Henning S. Mortveit, <a href="http://www.emis.de/journals/JACO/Volume33_1/hgv665924j44t770.html">Dynamics groups of asynchronous cellular automata</a>, Journal of Algebraic Combinatorics, Vol 33, No 1 (2011), pp. 11-35.

%H Souvik Roy, Nazim Fatès, and Sukanta Das, <a href="https://inria.hal.science/hal-04456320">Reversibility of Elementary Cellular Automata with fully asynchronous updating: an analysis of the rules with partial recurrence</a>, hal-04456320 [nlin.CG], [cs], 2024. See p. 17.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,2).

%F e=-1/2+i*sqrt(3)/2, e^2=-1/2-i*sqrt(3)/2, x=(1+sqrt(26/27))^(1/3)+(1-sqrt(26/27))^(1/3), y=e*(1+sqrt(26/27))^(1/3)+(e^2)*(1-sqrt(26/27))^(1/3), z=(e^2)*(1+sqrt(26/27))^(1/3)+e*(1-sqrt(26/27))^(1/3), a(n)=x^n+y^n+z^n.

%e a(10)=2*a(7)+a(8): 62=2*14+34.

%t LinearRecurrence[{0, 1, 2}, {3, 0, 2}, 50] (* _T. D. Noe_, Nov 05 2013 *)

%t Table[RootSum[-2 - #1 + #1^3 &, #^n &], {n, 0, 40}] (* _Eric W. Weisstein_, Dec 09 2014 *)

%Y Cf. A001608.

%K easy,nonn

%O 0,1

%A _Miklos Kristof_, Jul 15 2002

%E Deleted certain dangerous or potentially dangerous links. - _N. J. A. Sloane_, Jan 30 2021