

A077957


Powers of 2 alternating with zeros.


54



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
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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 selfavoiding 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 Ccurve at the nth iteration step. Therefore, L(n) is the Q(sqrt(2)) integer a(n) + a(n1)*sqrt(2), with a(1) = 0. For a variant of this Ccurve see A251732 and A251733.  Wolfdieter Lang, Dec 08 2014
a(n) counts walks (closed) on the graph G(1vertex,2loop,2loop). 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(n2) 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
Also the number of alternately constant compositions of n + 2, ranked by A351010. The alternately strict version gives A000213. The unordered version is A035363, ranked by A000290, strict A035457.  Gus Wiseman, Feb 19 2022


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/(12*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
a(n) = 2*a(n2) with a(0)=1, a(1)=0.  Jim Singh, Jul 12 2018


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 *)
LinearRecurrence[{0, 2}, {1, 0}, 54] (* Robert G. Wilson v, Jul 23 2018 *)
Riffle[2^Range[0, 30], 0, {2, 1, 2}] (* Harvey P. Dale, Jan 06 2022 *)


PROG

(PARI) a(n)=if(n<0n%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(); [next(a) for i in range(40)] # Peter Luschny, Jul 11 2013
(Magma) &cat [[2^n, 0]: n in [0..20]]; // Vincenzo Librandi, Apr 03 2018
(GAP) Flat(List([0..30], n>[2^n, 0])); # Muniru A Asiru, Aug 05 2018


CROSSREFS

Cf. A000079, A077966.
Column k=3 of A219946.  Alois P. Heinz, Dec 01 2012
Cf. A016116 (powers repeated).
Sequence in context: A349232 A194795 A131575 * A077966 A275670 A021102
Adjacent sequences: A077954 A077955 A077956 * A077958 A077959 A077960


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane, Nov 17 2002


STATUS

approved



