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
KEYWORD
nonn,easy
AUTHOR
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
EXTENSIONS
More terms from James A. Sellers, Jun 08 2000
STATUS
approved