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A004171 a(n) = 2^(2n+1). 62
2, 8, 32, 128, 512, 2048, 8192, 32768, 131072, 524288, 2097152, 8388608, 33554432, 134217728, 536870912, 2147483648, 8589934592, 34359738368, 137438953472, 549755813888, 2199023255552, 8796093022208, 35184372088832, 140737488355328, 562949953421312 (list; graph; refs; listen; history; text; internal format)
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 <=3 (0 is counted as part). (End)

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

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

Tanya Khovanova, Recursive Sequences

M. Paukner, L. Pepin, M. Riehl, and J. Wieser, Pattern Avoidance in Task-Precedence Posets, arXiv:1511.00080 [math.CO], 2015-2016.

Index to divisibility sequences

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 through infinity) 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) = A002063(n)/3 - A000302(n). - Zhandos Mambetaliyev, Nov 19 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

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 <=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 <=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^(2n + 1), {n, 0, 24}]

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

CROSSREFS

Cf. A013708-A013729.

Absolute value of A009117. Essentially the same as A081294.

Cf. A164632. Equals A000980(n) + 2*A181765(n). Cf. A013776.

Sequence in context: A274524 A081294 * A009117 A160637 A183895 A228921

Adjacent sequences:  A004168 A004169 A004170 * A004172 A004173 A004174

KEYWORD

easy,nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified August 19 14:06 EDT 2017. Contains 290808 sequences.