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 A052535 Expansion of (1-x)*(1+x)/(1-x-2*x^2+x^4). 6
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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 Number of tilings of a 4 X 2n rectangle by 4 X 1 tetrominoes. - M. Poyraz Torcuk, Dec 10 2021 LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 V. E. Hoggatt, Jr. and Marjorie Bicknell, Palindromic compositions, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356. 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). FORMULA 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 a(n) = A158943(n) -A158943(n-2). - R. J. Mathar, Jan 13 2023 MAPLE spec := [S, {S=Sequence(Prod(Z, Union(Z, Sequence(Prod(Z, Z)))))}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20); MATHEMATICA 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 *) PROG (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:=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 CROSSREFS Cf. A275446. Bisection of A003269 (odd part), Cf. A236582, A251074, A236580. Sequence in context: A024502 A280254 A280917 * A027988 A238859 A224960 Adjacent sequences: A052532 A052533 A052534 * A052536 A052537 A052538 KEYWORD easy,nonn AUTHOR encyclopedia(AT)pommard.inria.fr, Jan 25 2000 EXTENSIONS More terms from James A. Sellers, Jun 05 2000 STATUS approved

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Last modified December 1 21:17 EST 2023. Contains 367502 sequences. (Running on oeis4.)