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A052535
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Expansion of (1-x)*(1+x)/(1-x-2*x^2+x^4).
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6
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1, 1, 2, 4, 7, 14, 26, 50, 95, 181, 345, 657, 1252, 2385, 4544, 8657, 16493, 31422, 59864, 114051, 217286, 413966, 788674, 1502555, 2862617, 5453761, 10390321, 19795288, 37713313, 71850128, 136886433, 260791401, 496850954, 946583628
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OFFSET
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0,3
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COMMENTS
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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.
Diagonal sums of A054142. - Paul Barry, Jan 21 2005
Equals INVERT transform of (1, 1, 1, 0, 1, 0, 1, 0, 1,...). - Gary W. Adamson, Apr 27 2009
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REFERENCES
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V. E. Hoggatt, Jr., and Marjorie Bicknell, Palindromic compositions, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.
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LINKS
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G. C. Greubel, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 465
Todd Mullen, On Variants of Diffusion, Dalhousie University (Halifax, NS Canada, 2020).
Todd Mullen, Richard Nowakowski, and Danielle Cox, Counting Path Configurations in Parallel Diffusion, arXiv:2010.04750 [math.CO], 2020.
Index entries for linear recurrences with constant coefficients, signature (1,2,0,-1).
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FORMULA
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G.f.: (1 - x^2)/(1 - x - 2*x^2 + x^4).
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.
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).
a(n) = Sum_{k=0..floor(n/2)} binomial(2*n-3*k, k). - Paul Barry, Jan 21 2005
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MAPLE
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spec := [S, {S=Sequence(Prod(Z, Union(Z, Sequence(Prod(Z, Z)))))}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);
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MATHEMATICA
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CoefficientList[Series[(1-x^2)/(1-x-2x^2+x^4), {x, 0, 40}], x] (* or *)
Table[Length@ Flatten[Map[Permutations, DeleteCases[IntegerPartitions@ n, {___, a_, ___} /; And[EvenQ@ a, a != 2]]], 1], {n, 0, 40}] (* Michael De Vlieger, Aug 17 2016 *)
LinearRecurrence[{1, 2, 0, -1}, {1, 1, 2, 4}, 40] (* Harvey P. Dale, Apr 12 2018 *)
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PROG
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(PARI) my(x='x+O('x^40)); Vec((1-x^2)/(1-x-2*x^2+x^4)) \\ G. C. Greubel, May 09 2019
(MAGMA) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x^2)/( 1-x-2*x^2+x^4) )); // G. C. Greubel, May 09 2019
(Sage) ((1-x^2)/(1-x-2*x^2+x^4)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, May 09 2019
(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
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CROSSREFS
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Cf. A275446.
Sequence in context: A024502 A280254 A280917 * A027988 A238859 A224960
Adjacent sequences: A052532 A052533 A052534 * A052536 A052537 A052538
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KEYWORD
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easy,nonn
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AUTHOR
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encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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EXTENSIONS
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More terms from James A. Sellers, Jun 05 2000
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STATUS
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approved
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