A052938 Expansion of (1 + 2*x - 2*x^2)/( (1+x)*(1-x)^2 ).

1, 3, 2, 4, 3, 5, 4, 6, 5, 7, 6, 8, 7, 9, 8, 10, 9, 11, 10, 12, 11, 13, 12, 14, 13, 15, 14, 16, 15, 17, 16, 18, 17, 19, 18, 20, 19, 21, 20, 22, 21, 23, 22, 24, 23, 25, 24, 26, 25, 27, 26, 28, 27, 29, 28, 30, 29, 31, 30, 32, 31, 33, 32, 34, 33, 35, 34, 36, 35, 37, 36, 38, 37, 39

Offset

0,2

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 929

Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Formula

G.f.: (1+2*x-2*x^2)/((1+x)*(1-x)^2).

a(n) = -a(n-1) + n + 3, with a(0)=1.

a(n) = (3*(-1)^(n+1) + 2*n + 7)/4.

A112034(n) = 3*2^a(n); a(n) = A109613(n+2) - A084964(n). - Reinhard Zumkeller, Aug 27 2005

a(n) = A035106(n+3) - A035106(n+2). - Reinhard Zumkeller, Oct 06 2015

a(n) = A060762(n+1) - 1. - Filip Zaludek, Nov 19 2016

E.g.f.: ((5+x)*sinh(x) + (2+x)*cosh(x))/2. - G. C. Greubel, Oct 18 2019

Maple

spec := [S, {S=Prod(Union(Sequence(Z), Z, Z), Sequence(Prod(Z, Z)))}, unlabeled ]: seq(combstruct[count ](spec, size=n), n=0..20);
seq((2*n+7-3*(-1)^n)/4, n=0..30); # G. C. Greubel, Oct 18 2019

Mathematica

LinearRecurrence[{1, 1, -1}, {1, 3, 2}, 80] (* Harvey P. Dale, Apr 10 2019 *)

Prog

(PARI) a(n)=([0, 1, 0; 0, 0, 1; -1, 1, 1]^n*[1; 3; 2])[1, 1] \\ Charles R Greathouse IV, Sep 02 2015
(Haskell)
a052938 n = a052938_list !! n
a052938_list = 1 : 3 : 2 : zipWith (-) [5..] a052938_list
-- Reinhard Zumkeller, Oct 06 2015
(Magma) [(2*n+7-3*(-1)^n)/4: n in [0..30]]; // G. C. Greubel, Oct 18 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 75); Coefficients(R!( (1 + 2*x - 2*x^2)/( (1+x)*(1-x)^2 ))); // Marius A. Burtea, Oct 18 2019
(Sage) [(2*n+7-3*(-1)^n)/4 for n in (0..30)] # G. C. Greubel, Oct 18 2019
(GAP) List([0..30], n-> (2*n+7-3*(-1)^n)/4); # G. C. Greubel, Oct 18 2019

Crossrefs

Cf. A028242 (same sequence with 1,0,2 prefix).

Cf. A035106.

Sequence in context: A119707 A354679 A307118 * A302391 A140114 A243852

Adjacent sequences: A052935 A052936 A052937 * A052939 A052940 A052941

Keyword

easy,nonn

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Extensions

More terms from James A. Sellers, Jun 06 2000

Status

approved