%I #131 Dec 19 2023 17:45:01

%S 2,8,32,128,512,2048,8192,32768,131072,524288,2097152,8388608,

%T 33554432,134217728,536870912,2147483648,8589934592,34359738368,

%U 137438953472,549755813888,2199023255552,8796093022208,35184372088832,140737488355328,562949953421312

%N a(n) = 2^(2n+1).

%C Same as Pisot sequences E(2, 8), L(2, 8), P(2, 8), T(2, 8). See A008776 for definitions of Pisot sequences.

%C In the Chebyshev polynomial of degree 2n, a(n) is the coefficient of x^2n. - _Benoit Cloitre_, Mar 13 2002

%C 1/2 - 1/8 + 1/32 - 1/128 + ... = 2/5. - _Gary W. Adamson_, Mar 03 2009

%C From _Adi Dani_, May 15 2011: (Start)

%C Number of ways of placing an even number of indistinguishable objects in n + 1 distinguishable boxes with at most 3 objects in box.

%C Number of compositions of even natural numbers into n + 1 parts less than or equal to 3 (0 is counted as part). (End)

%C Also the number of maximal cliques in the (n+1)-Sierpinski tetrahedron graph for n > 0. - _Eric W. Weisstein_, Dec 01 2017

%C 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

%C 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

%D Adi Dani, Quasicompositions of natural numbers, Proceedings of III congress of mathematicians of Macedonia, Struga Macedonia 29 IX -2 X 2005 pages 225-238.

%H Vincenzo Librandi, <a href="/A004171/b004171.txt">Table of n, a(n) for n = 0..200</a>

%H J. Brillhart and P. Morton, <a href="http://www.maa.org/programs/maa-awards/writing-awards/a-case-study-in-mathematical-research-the-golay-rudin-shapiro-sequence">A case study in mathematical research: the Golay-Rudin-Shapiro sequence</a>, Amer. Math. Monthly, 103 (1996) 854-869.

%H Ling Gao, <a href="http://hdl.handle.net/20.500.12680/h989rb533">Graph assembly for spider and tadpole graphs</a>, Master's Thesis, Cal. State Poly. Univ. (2023).

%H Milan Janjic, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfun.pdf">Enumerative Formulas for Some Functions on Finite Sets</a>.

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>.

%H Mitchell Paukner, Lucy Pepin, Manda Riehl, and Jarred Wieser, <a href="https://arxiv.org/abs/1511.00080">Pattern Avoidance in Task-Precedence Posets</a>, arXiv:1511.00080 [math.CO], 2015-2016.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MaximalClique.html">Maximal Clique</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SierpinskiTetrahedronGraph.html">Sierpinski Tetrahedron Graph</a>.

%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (4).

%H <a href="/index/Di#divseq">Index to divisibility sequences</a>.

%F a(n) = 2*4^n.

%F a(n) = 4*a(n-1).

%F 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

%F From _Philippe Deléham_, Nov 23 2008: (Start)

%F a(n) = 2*A000302(n).

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

%F 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

%F E.g.f.: 2*exp(4*x). - _Ilya Gutkovskiy_, Nov 01 2016

%F a(n) = A002063(n)/3 - A000302(n). - _Zhandos Mambetaliyev_, Nov 19 2016

%F 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

%F Product_{n>=0} (1 - 1/a(n)) = A132020. - _Amiram Eldar_, May 08 2023

%e G.f. = 2 + 8*x + 32*x^2 + 128*x^3 + 512*x^4 + 2048*x^5 + 8192*x^6 + 32768*x^7 + ...

%e From _Adi Dani_, May 15 2011: (Start)

%e a(1) = 8 because all compositions of even natural numbers into 2 parts less than or equal to 3 are:

%e for 0: (0, 0)

%e for 2: (0, 2), (2, 0), (1, 1)

%e for 4: (1, 3), (3, 1), (2, 2)

%e for 6: (3, 3).

%e a(2) = 32 because all compositions of even natural numbers into 3 parts less than or equal to 3 are:

%e for 0: (0, 0, 0)

%e for 2: (0, 0, 2), (0, 2, 0), (2, 0, 0), (0, 1, 1), (1, 0, 1) , (1, 1, 0)

%e 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)

%e 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)

%e for 8: (2, 3, 3), (3, 2, 3), (3, 3, 2).

%e (End)

%p seq(2^(2*n+1),n=0..24); # _Nathaniel Johnston_, Jun 25 2011

%t Table[2^(2 n + 1), {n, 0, 24}]

%t 2^(2 Range[20] - 1) (* _Eric W. Weisstein_, Dec 01 2017 *)

%t LinearRecurrence[{4}, {2}, 20] (* _Eric W. Weisstein_, Dec 01 2017 *)

%t CoefficientList[Series[2/(1 - 4 x), {x, 0, 20}], x] (* _Eric W. Weisstein_, Dec 01 2017 *)

%o (Magma) [2^(2*n+1): n in [0..30]]; // _Vincenzo Librandi_, May 16 2011

%o (PARI) a(n)=2<<(2*n) \\ _Charles R Greathouse IV_, Apr 07 2012

%o (PARI) a(n) = 2^(2*n+1) \\ _Michel Marcus_, Aug 12 2014

%o (Haskell)

%o a004171 = (* 2) . a000302

%o a004171_list = iterate (* 4) 2 -- _Reinhard Zumkeller_, Jan 09 2013

%o (GAP) List([0..30],n->2^(2*n+1)); # _Muniru A Asiru_, Mar 12 2019

%o (Scala) ((List.fill(20)(4: BigInt)).scanLeft(1: BigInt)(_ * _)).map(2 * _) // _Alonso del Arte_, Sep 12 2019

%Y Cf. A013708-A013729.

%Y Absolute value of A009117. Essentially the same as A081294.

%Y Cf. A132020, A164632. Equals A000980(n) + 2*A181765(n). Cf. A013776.

%K easy,nonn

%O 0,1

%A _N. J. A. Sloane_