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A052535 Expansion of (1-x)*(1+x)/(1-x-2*x^2+x^4).

%I A052535 #66 Jul 02 2025 16:01:58
%S A052535 1,1,2,4,7,14,26,50,95,181,345,657,1252,2385,4544,8657,16493,31422,
%T A052535 59864,114051,217286,413966,788674,1502555,2862617,5453761,10390321,
%U A052535 19795288,37713313,71850128,136886433,260791401,496850954,946583628
%N A052535 Expansion of (1-x)*(1+x)/(1-x-2*x^2+x^4).
%C A052535 a(n) = number of compositions of n with parts in {2,1,3,5,7,9,...}. The generating function follows easily from Theorem 1.1 of the Hoggatt et al. reference. Example: a(4)= 7 because we have 22, 31, 13, 211, 121, 112, and 1111. - _Emeric Deutsch_, Aug 17 2016.
%C A052535 Diagonal sums of A054142. - _Paul Barry_, Jan 21 2005
%C A052535 Equals INVERT transform of (1, 1, 1, 0, 1, 0, 1, 0, 1, ...). - _Gary W. Adamson_, Apr 27 2009
%C A052535 Number of tilings of a 4 X 2n rectangle by 4 X 1 tetrominoes. - _M. Poyraz Torcuk_, Dec 10 2021
%H A052535 G. C. Greubel, <a href="/A052535/b052535.txt">Table of n, a(n) for n = 0..1000</a>
%H A052535 Jean-Luc Baril, Nathanaël Hassler, Sergey Kirgizov, and José L. Ramírez, <a href="https://arxiv.org/abs/2402.04851">Grand zigzag knight's paths</a>, arXiv:2402.04851 [math.CO], 2024. See p. 18.
%H A052535 V. E. Hoggatt, Jr. and Marjorie Bicknell, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/13-4/hoggatt1.pdf">Palindromic compositions</a>, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.
%H A052535 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=465">Encyclopedia of Combinatorial Structures 465</a>
%H A052535 Todd Mullen, <a href="https://dalspace.library.dal.ca/bitstream/handle/10222/78458/Mullen-Todd-PhD-MATH-April-2020.pdf">On Variants of Diffusion</a>, Dalhousie University (Halifax, NS Canada, 2020).
%H A052535 Todd Mullen, Richard Nowakowski, and Danielle Cox, <a href="https://arxiv.org/abs/2010.04750">Counting Path Configurations in Parallel Diffusion</a>, arXiv:2010.04750 [math.CO], 2020.
%H A052535 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,0,-1).
%F A052535 G.f.: (1 - x^2)/(1 - x - 2*x^2 + x^4).
%F A052535 a(n) = a(n-1) + 2*a(n-2) - a(n-4), with a(0)=1, a(1)=1, a(2)=2, a(3)=4.
%F A052535 a(n) = Sum_{alpha = RootOf(1-x-2*x^2+x^4)} (1/283)*(27 + 112*alpha + 9*alpha^2 -48*alpha^3)*alpha^(-n-1).
%F A052535 a(n) = Sum_{k=0..floor(n/2)} binomial(2*n-3*k, k). - _Paul Barry_, Jan 21 2005
%F A052535 a(n) = A158943(n) -A158943(n-2). - _R. J. Mathar_, Jan 13 2023
%p A052535 spec := [S,{S=Sequence(Prod(Z,Union(Z,Sequence(Prod(Z,Z)))))},unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);
%t A052535 CoefficientList[Series[(1-x^2)/(1-x-2x^2+x^4), {x, 0, 40}], x] (* or *)
%t A052535 Table[Length@ Flatten[Map[Permutations, DeleteCases[IntegerPartitions@ n, {___, a_, ___} /; And[EvenQ@ a, a != 2]]], 1], {n, 0, 40}]  (* _Michael De Vlieger_, Aug 17 2016 *)
%t A052535 LinearRecurrence[{1,2,0,-1},{1,1,2,4},40] (* _Harvey P. Dale_, Apr 12 2018 *)
%o A052535 (PARI) my(x='x+O('x^40)); Vec((1-x^2)/(1-x-2*x^2+x^4)) \\ _G. C. Greubel_, May 09 2019
%o A052535 (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x^2)/( 1-x-2*x^2+x^4) )); // _G. C. Greubel_, May 09 2019
%o A052535 (Sage) ((1-x^2)/(1-x-2*x^2+x^4)).series(x, 40).coefficients(x, sparse=False) # _G. C. Greubel_, May 09 2019
%o A052535 (GAP) a:=[1,1,2,4];; for n in [5..40] do a[n]:=a[n-1]+2*a[n-2]-a[n-4]; od; a; # _G. C. Greubel_, May 09 2019
%Y A052535 Cf. A275446.
%Y A052535 Bisection of A003269 (odd part),
%Y A052535 Cf. A236582, A251074, A236580.
%K A052535 easy,nonn
%O A052535 0,3
%A A052535 encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E A052535 More terms from _James Sellers_, Jun 05 2000