OFFSET
0,1
COMMENTS
Same as Pisot sequences E(2, 8), L(2, 8), P(2, 8), T(2, 8). See A008776 for definitions of Pisot sequences.
In the Chebyshev polynomial of degree 2n, a(n) is the coefficient of x^2n. - Benoit Cloitre, Mar 13 2002
1/2 - 1/8 + 1/32 - 1/128 + ... = 2/5. - Gary W. Adamson, Mar 03 2009
From Adi Dani, May 15 2011: (Start)
Number of ways of placing an even number of indistinguishable objects in n + 1 distinguishable boxes with at most 3 objects in box.
Number of compositions of even natural numbers into n + 1 parts less than or equal to 3 (0 is counted as part). (End)
Also the number of maximal cliques in the (n+1)-Sierpinski tetrahedron graph for n > 0. - Eric W. Weisstein, Dec 01 2017
Assuming the Collatz conjecture is true, any starting number eventually leads to a power of 2. A number in this sequence can never be the first power of 2 in a Collatz sequence except of course for the Collatz sequence starting with that number. For example, except for 8, 4, 2, 1, any Collatz sequence that includes 8 must also include 16 (e.g., 5, 16, 8, 4, 2, 1). - Alonso del Arte, Oct 01 2019
First differences of A020988, and thus the "wavelengths" of the local maxima in A020986. See the Brillhart and Morton link, pp. 855-856. - John Keith, Mar 04 2021
REFERENCES
Adi Dani, Quasicompositions of natural numbers, Proceedings of III congress of mathematicians of Macedonia, Struga Macedonia 29 IX -2 X 2005 pages 225-238.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
J. Brillhart and P. Morton, A case study in mathematical research: the Golay-Rudin-Shapiro sequence, Amer. Math. Monthly, 103 (1996) 854-869.
Ling Gao, Graph assembly for spider and tadpole graphs, Master's Thesis, Cal. State Poly. Univ. (2023).
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
Tanya Khovanova, Recursive Sequences.
Mitchell Paukner, Lucy Pepin, Manda Riehl, and Jarred Wieser, Pattern Avoidance in Task-Precedence Posets, arXiv:1511.00080 [math.CO], 2015-2016.
Eric Weisstein's World of Mathematics, Maximal Clique.
Eric Weisstein's World of Mathematics, Sierpinski Tetrahedron Graph.
Index entries for linear recurrences with constant coefficients, signature (4).
FORMULA
a(n) = 2*4^n.
a(n) = 4*a(n-1).
1 = 1/2 + Sum_{n >= 1} 3/a(n) = 3/6 + 3/8 + 3/32 + 3/128 + 3/512 + 3/2048 + ...; with partial sums: 1/2, 31/32, 127/128, 511/512, 2047/2048, ... - Gary W. Adamson, Jun 16 2003
From Philippe Deléham, Nov 23 2008: (Start)
a(n) = 2*A000302(n).
G.f.: 2/(1-4*x). (End)
a(n) = A081294(n+1) = A028403(n+1) - A000079(n+1) for n >= 1. a(n-1) = A028403(n) - A000079(n). - Jaroslav Krizek, Jul 27 2009
E.g.f.: 2*exp(4*x). - Ilya Gutkovskiy, Nov 01 2016
a(n) = Sum_{k = 0..2*n} (-1)^(k+n)*binomial(4*n + 2, 2*k + 1); a(2*n) = Sum_{k = 0..2*n} binomial(4*n + 2, 2*k + 1) = A013776(n). - Peter Bala, Nov 25 2016
Product_{n>=0} (1 - 1/a(n)) = A132020. - Amiram Eldar, May 08 2023
EXAMPLE
G.f. = 2 + 8*x + 32*x^2 + 128*x^3 + 512*x^4 + 2048*x^5 + 8192*x^6 + 32768*x^7 + ...
From Adi Dani, May 15 2011: (Start)
a(1) = 8 because all compositions of even natural numbers into 2 parts less than or equal to 3 are:
for 0: (0, 0)
for 2: (0, 2), (2, 0), (1, 1)
for 4: (1, 3), (3, 1), (2, 2)
for 6: (3, 3).
a(2) = 32 because all compositions of even natural numbers into 3 parts less than or equal to 3 are:
for 0: (0, 0, 0)
for 2: (0, 0, 2), (0, 2, 0), (2, 0, 0), (0, 1, 1), (1, 0, 1) , (1, 1, 0)
for 4: (0, 1, 3), (0, 3, 1), (1, 0, 3), (1, 3, 0), (3, 0, 1), (3, 1, 0), (0, 2, 2), (2, 0, 2), (2, 2, 0), (1, 1, 2), (1, 2, 1), (2, 1, 1)
for 6: (0, 3, 3), (3, 0, 3), (3, 3, 0), (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1), (2, 2, 2)
for 8: (2, 3, 3), (3, 2, 3), (3, 3, 2).
(End)
MAPLE
seq(2^(2*n+1), n=0..24); # Nathaniel Johnston, Jun 25 2011
MATHEMATICA
Table[2^(2 n + 1), {n, 0, 24}]
2^(2 Range[20] - 1) (* Eric W. Weisstein, Dec 01 2017 *)
LinearRecurrence[{4}, {2}, 20] (* Eric W. Weisstein, Dec 01 2017 *)
CoefficientList[Series[2/(1 - 4 x), {x, 0, 20}], x] (* Eric W. Weisstein, Dec 01 2017 *)
PROG
(Magma) [2^(2*n+1): n in [0..30]]; // Vincenzo Librandi, May 16 2011
(PARI) a(n)=2<<(2*n) \\ Charles R Greathouse IV, Apr 07 2012
(PARI) a(n) = 2^(2*n+1) \\ Michel Marcus, Aug 12 2014
(Haskell)
a004171 = (* 2) . a000302
a004171_list = iterate (* 4) 2 -- Reinhard Zumkeller, Jan 09 2013
(GAP) List([0..30], n->2^(2*n+1)); # Muniru A Asiru, Mar 12 2019
(Scala) ((List.fill(20)(4: BigInt)).scanLeft(1: BigInt)(_ * _)).map(2 * _) // Alonso del Arte, Sep 12 2019
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
STATUS
approved