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A077957 Powers of 2 alternating with zeros. 43
1, 0, 2, 0, 4, 0, 8, 0, 16, 0, 32, 0, 64, 0, 128, 0, 256, 0, 512, 0, 1024, 0, 2048, 0, 4096, 0, 8192, 0, 16384, 0, 32768, 0, 65536, 0, 131072, 0, 262144, 0, 524288, 0, 1048576, 0, 2097152, 0, 4194304, 0, 8388608, 0, 16777216, 0, 33554432, 0, 67108864, 0, 134217728, 0, 268435456 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Normally sequences like this are not included, since with the alternating 0's deleted it is already in the database.

Inverse binomial transform of A001333. - Paul Barry, Feb 25 2003

"Sloping binary representation" of powers of 2 (A000079), slope=-1 (see A037095 and A102370). - Philippe Deléham, Jan 04 2008

0,1,0,2,0,4,0,8,0,16,...is the inverse binomial transform of A000129 (Pell numbers). - Philippe Deléham, Oct 28 2008

Number of maximal self-avoiding walks from the NW to SW corners of a 3 X n grid.

Row sums of the triangle in A204293. - Reinhard Zumkeller, Jan 14 2012

Pisano period lengths: 1, 1, 4, 1, 8, 4, 6, 1, 12, 8, 20, 4, 24, 6, 8, 1, 16, 12, 36, 8, ... . - R. J. Mathar, Aug 10 2012

This sequence occurs in the length L(n) = sqrt(2)^n of Lévy's C-curve at the n-th iteration step. Therefore, L(n) is the Q(sqrt(2)) integer a(n) + a(n-1)*sqrt(2), with a(-1) = 0. For a variant of this C-curve see A251732 and A51733. - Wolfdieter Lang, Dec 08 2014

a(n) counts walks (closed) on the graph G(1-vertex,2-loop,2-loop). Equivalently the middle entry (2,2) of A^n where the adjacency matrix of digraph is A=(0,1,0;1,0,1;0,1,0). - David Neil McGrath, Dec 19 2014

a(n-2) is the number of compositions of n into even parts. For example, there are 4 compositions of 6 into even parts: (6), (222), (42), and (24). - David Neil McGrath, Dec 19 2014

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (0,2).

FORMULA

G.f.: 1/(1-2*x^2).

E.g.f.: cosh(x*sqrt(2)).

a(n) = (1 - n mod 2) * 2^floor(n/2).

a(n) = sqrt(2)^n*(1+(-1)^n)/2. - Paul Barry, May 13 2003

Construct the power matrix T(n,j)=[A^*j]*[S^*(j-1)] with A=(0,1,0,1,...) and S=(0,1,0,0...) or A063524. [* is convolution operator]. Define S^*0=I with I=(1,0,0...). Then a(n-2)=sum[j=1...n,T(n,j)]. - David Neil McGrath, Dec 22 2014

MAPLE

seq(op([2^n, 0]), n=0..100); # Robert Israel, Dec 23 2014

MATHEMATICA

a077957[n_] := Riffle[Table[2^i, {i, 0, n - 1}], Table[0, {n}]]; a077957[29] (* Michael De Vlieger, Dec 22 2014 *)

CoefficientList[Series[1/(1 - 2*x^2), {x, 0, 50}], x] (* G. C. Greubel, Apr 12 2017 *)

PROG

(PARI) a(n)=if(n<0||n%2, 0, 2^(n/2))

(Haskell)

a077957 = sum . a204293_row  -- Reinhard Zumkeller, Jan 14 2012

(Sage)

def A077957():

    x, y = -1, 1

    while true:

        yield -x

        x, y = x + y, x - y

a = A077957(); [a.next() for i in range(40)]  # Peter Luschny, Jul 11 2013

CROSSREFS

Cf. A000079, A077966.

Column k=3 of A219946. - Alois P. Heinz, Dec 01 2012

Sequence in context: A194795 A131575 * A077966 A275670 A021102 A021053

Adjacent sequences:  A077954 A077955 A077956 * A077958 A077959 A077960

KEYWORD

nonn,easy,changed

AUTHOR

N. J. A. Sloane, Nov 17 2002

STATUS

approved

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Last modified April 26 15:40 EDT 2017. Contains 285446 sequences.