A052992 - OEIS

%I #74 Jan 05 2025 00:45:16

%S 1,1,5,5,21,21,85,85,341,341,1365,1365,5461,5461,21845,21845,87381,

%T 87381,349525,349525,1398101,1398101,5592405,5592405,22369621,

%U 22369621,89478485,89478485,357913941,357913941,1431655765,1431655765,5726623061,5726623061

%N Expansion of 1/((1 - x)*(1 - 2*x)*(1 + 2*x)).

%C a(n) is the sum of square divisors of 2^n. - _Paul Barry_, Oct 13 2005

%C Decimal representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 131", based on the 5-celled von Neumann neighborhood.  See A279053 for references and links. - _Robert Price_, Dec 05 2016

%H Vincenzo Librandi, <a href="/A052992/b052992.txt">Table of n, a(n) for n = 0..1000</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=1068">Encyclopedia of Combinatorial Structures 1068</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,4,-4).

%F G.f.: 1/(-1+4*x^2)/(-1+x).

%F Recurrence: {a(1)=1, a(0)=1, -4*a(n) - 1 + a(n+2) = 0}.

%F a(n) = -1/3 + Sum((1/6)*(1+4*_alpha)*_alpha^(-1-n), where _alpha=RootOf(-1+4*_Z^2))

%F a(n) = Sum_{k=0..n} 2^k(1+(-1)^k)/2. - _Paul Barry_, Nov 24 2003

%F a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3). - _Paul Curtz_, Apr 27 2011

%F a(n) = (4^(1 + floor(n/2)) - 1)/3. - _Federico Provvedi_, Oct 19 2018

%F a(n)-a(n-1) = A199572(n). - _R. J. Mathar_, Feb 27 2019

%F a(n) = A263053(n)/2. - _Pascal Bisson_, Feb 03 2022

%p spec := [S,{S=Prod(Sequence(Prod(Union(Z,Z),Union(Z,Z))),Sequence(Z))},unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);

%t CoefficientList[Series[1/((1-x)(1-2x)(1+2x)),{x,0,40}],x] (* or *) LinearRecurrence[{1,4,-4},{1,1,5},40] (* or *) With[{c= LinearRecurrence[ {5,-4},{1,5},20]},Riffle[c,c]] (* _Harvey P. Dale_, Sep 12 2015 *)

%t (4^(1 + Floor[(Range@40-1)/2])-1)/3 (* _Federico Provvedi_, Oct 19 2018 *)

%o (Python)

%o for n in range(0,40): print((int(4**(1+int((n+2)/2)-1)/3)), end=', ') # _Stefano Spezia_, Oct 19 2018

%o (Python) [4**(1+(n+2)//2-1)//3 for n in range(40)] # _Pascal Bisson_, Feb 03 2022

%o (GAP) Flat(List([1..17],n->[(4^n-1)/3,(4^n-1)/3])); # _Muniru A Asiru_, Oct 21 2018

%o (Magma) [&+[2^k*(1 + (-1)^k)/2: k in [0..n]]: n in [0..50]]; // _Vincenzo Librandi_, Oct 21 2018

%Y Cf. A263053, A279053.

%K nonn,easy

%O 0,3

%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000

%E More terms from _James A. Sellers_, Jun 08 2000