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 A052535 Expansion of (1-x)(1+x)/(1-x-2x^2+x^4). 5
 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,...). [From Gary W. Adamson, Apr 27 2009] REFERENCES V. E. Hoggatt, Jr., and Marjorie Bicknell, Palindromic compositions, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356. LINKS INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 465 Index entries for linear recurrences with constant coefficients, signature (1,2,0,-1). FORMULA G.f.: -(-1+x^2)/(1-2*x^2+x^4-x) Recurrence: {a(1)=1, a(0)=1, a(3)=4, a(2)=2, a(n)-2*a(n+2)-a(n+3)+a(n+4)=0} Sum(-1/283*(-112*_alpha+48*_alpha^3-9*_alpha^2-27)*_alpha^(-1-n), _alpha=RootOf(1-2*_Z^2+_Z^4-_Z)) a(n)=sum{k=0..floor(n/2), binomial(2n-3k, k)}. - Paul Barry, Jan 21 2005 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 - 2 x^2 + x^4 - x), {x, 0, 33}], x] (* or *) Table[Length@ Flatten[Map[Permutations, DeleteCases[IntegerPartitions@ n, {___, a_, ___} /; And[EvenQ@ a, a != 2]]], 1], {n, 0, 27}]  (* Michael De Vlieger, Aug 17 2016 *) CROSSREFS Cf. A275446. 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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