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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

REFERENCES

V. E. Hoggatt, Jr., and Marjorie Bicknell, Palindromic compositions, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

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 - 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

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<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

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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Last modified July 14 16:05 EDT 2020. Contains 335729 sequences. (Running on oeis4.)