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%I #134 Mar 24 2024 13:17:58
%S 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,
%T 0,8192,0,16384,0,32768,0,65536,0,131072,0,262144,0,524288,0,1048576,
%U 0,2097152,0,4194304,0,8388608,0,16777216,0,33554432,0,67108864,0,134217728,0,268435456
%N Powers of 2 alternating with zeros.
%C Normally sequences like this are not included, since with the alternating 0's deleted it is already in the database.
%C Inverse binomial transform of A001333. - _Paul Barry_, Feb 25 2003
%C "Sloping binary representation" of powers of 2 (A000079), slope=-1 (see A037095 and A102370). - _Philippe Deléham_, Jan 04 2008
%C 0,1,0,2,0,4,0,8,0,16,... is the inverse binomial transform of A000129 (Pell numbers). - _Philippe Deléham_, Oct 28 2008
%C Number of maximal self-avoiding walks from the NW to SW corners of a 3 X n grid.
%C Row sums of the triangle in A204293. - _Reinhard Zumkeller_, Jan 14 2012
%C 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
%C 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 A251733. - _Wolfdieter Lang_, Dec 08 2014
%C 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
%C 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
%C 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
%C a(n) counts degree n fixed points of GF(2)[x]'s automorphisms. Proof: given a field k, k[x]'s automorphisms are determined by k's automorphisms and invertible affine maps x -> ax + b. GF(2) is rigid and has only one unit so its only nontrivial automorphism is x -> x + 1. For n = 0 we have 1 fixed point, the constant polynomial 1. (Taking the convention that 0 is not a degree 0 polynomial.) For n = 1 we have 0 fixed points as x -> x + 1 -> x are the only degree 1 polynomials. Note that if f(x) is a fixed point, then f(x) + 1 is also a fixed point. Given f(x) a degree n fixed point, we can assume WLOG x | f(x). Applying the automorphism, we then have x + 1 | f(x). Now note that f(x) / (x^2 + x) must be a fixed point, so any fixed point of degree n must either be of the form g(x) * (x^2 + x) or g(x) * (x^2 + x) + 1 for a unique degree n - 2 fixed point g(x). Therefore we have the recurrence relation a(n) = 2 * a(n - 2) as desired. - _Keith J. Bauer_, Mar 19 2024
%H G. C. Greubel, <a href="/A077957/b077957.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,2).
%F G.f.: 1/(1-2*x^2).
%F E.g.f.: cosh(x*sqrt(2)).
%F a(n) = (1 - n mod 2) * 2^floor(n/2).
%F a(n) = sqrt(2)^n*(1+(-1)^n)/2. - _Paul Barry_, May 13 2003
%F a(n) = 2*a(n-2) with a(0)=1, a(1)=0. - _Jim Singh_, Jul 12 2018
%p seq(op([2^n,0]),n=0..100); # _Robert Israel_, Dec 23 2014
%t a077957[n_] := Riffle[Table[2^i, {i, 0, n - 1}], Table[0, {n}]]; a077957[29] (* _Michael De Vlieger_, Dec 22 2014 *)
%t CoefficientList[Series[1/(1 - 2*x^2), {x,0,50}], x] (* _G. C. Greubel_, Apr 12 2017 *)
%t LinearRecurrence[{0, 2}, {1, 0}, 54] (* _Robert G. Wilson v_, Jul 23 2018 *)
%t Riffle[2^Range[0,30],0,{2,-1,2}] (* _Harvey P. Dale_, Jan 06 2022 *)
%o (PARI) a(n)=if(n<0||n%2, 0, 2^(n/2))
%o (Haskell)
%o a077957 = sum . a204293_row -- _Reinhard Zumkeller_, Jan 14 2012
%o (Sage)
%o def A077957():
%o x, y = -1, 1
%o while True:
%o yield -x
%o x, y = x + y, x - y
%o a = A077957(); [next(a) for i in range(40)] # _Peter Luschny_, Jul 11 2013
%o (Magma) &cat [[2^n,0]: n in [0..20]]; // _Vincenzo Librandi_, Apr 03 2018
%o (GAP) Flat(List([0..30],n->[2^n,0])); # _Muniru A Asiru_, Aug 05 2018
%Y Cf. A000079, A077966.
%Y Column k=3 of A219946. - _Alois P. Heinz_, Dec 01 2012
%Y Cf. A016116 (powers repeated).
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_, Nov 17 2002
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