login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A052992 Expansion of 1/((1 - x)*(1 - 2*x)*(1 + 2*x)). 8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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).

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-1)/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

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 *)

PROG

(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

(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

CROSSREFS

Cf. A263053, A279053.

Sequence in context: A116400 A279810 A279750 * A280147 A279666 A283046

Adjacent sequences: A052989 A052990 A052991 * A052993 A052994 A052995

KEYWORD

nonn,easy

AUTHOR

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

EXTENSIONS

More terms from James A. Sellers, Jun 08 2000

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 30 01:31 EST 2022. Contains 358431 sequences. (Running on oeis4.)