|
|
A004171
|
|
a(n) = 2^(2n+1).
|
|
122
|
|
|
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
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
|
|
|
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
G.f.: 2/(1-4*x). (End)
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 + ...
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
|
|
|
MATHEMATICA
|
Table[2^(2 n + 1), {n, 0, 24}]
CoefficientList[Series[2/(1 - 4 x), {x, 0, 20}], x] (* Eric W. Weisstein, Dec 01 2017 *)
|
|
PROG
|
(Haskell)
a004171 = (* 2) . a000302
(Scala) ((List.fill(20)(4: BigInt)).scanLeft(1: BigInt)(_ * _)).map(2 * _) // Alonso del Arte, Sep 12 2019
|
|
CROSSREFS
|
|
|
KEYWORD
|
easy,nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|