A052992 Expansion of 1/((1 - x)*(1 - 2*x)*(1 + 2*x)).

1, 1, 5, 5, 21, 21, 85, 85, 341, 341, 1365, 1365, 5461, 5461, 21845, 21845, 87381, 87381, 349525, 349525, 1398101, 1398101, 5592405, 5592405, 22369621, 22369621, 89478485, 89478485, 357913941, 357913941, 1431655765, 1431655765, 5726623061, 5726623061

Offset: 0

Author

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

Keywords

• nonn
• easy

Comments

a(n) is the sum of square divisors of 2^n. - Paul Barry, Oct 13 2005

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1068
• Index entries for linear recurrences with constant coefficients, signature (1,4,-4).

Crossrefs

Cf. A263053, A279053.

Programs

GAP

Flat(List([1..17],n->[(4^n-1)/3,(4^n-1)/3])); # Muniru A Asiru, Oct 21 2018

Magma

[&+[2^k*(1 + (-1)^k)/2: k in [0..n]]: n in [0..50]]; // Vincenzo Librandi, Oct 21 2018

Maple

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

Mathematica

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 *)
(4^(1 + Floor[(Range@40-1)/2])-1)/3 (* Federico Provvedi, Oct 19 2018 *)

Python

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

Python

[4**(1+(n+2)//2-1)//3 for n in range(40)] # Pascal Bisson, Feb 03 2022

Formula

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

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

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

a(n) = Sum_{k=0..n} 2^k(1+(-1)^k)/2. - Paul Barry, Nov 24 2003

a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3). - Paul Curtz, Apr 27 2011

a(n) = (4^(1 + floor(n/2)) - 1)/3. - Federico Provvedi, Oct 19 2018

a(n)-a(n-1) = A199572(n). - R. J. Mathar, Feb 27 2019

a(n) = A263053(n)/2. - Pascal Bisson, Feb 03 2022

Extensions

More terms from James Sellers, Jun 08 2000