A071100 Expansion of (5 + 3*x + x^2 - x^3) / (1 - 2*x - 2*x^2 - 2*x^3 + x^4) in powers of x.

5, 13, 37, 109, 313, 905, 2617, 7561, 21853, 63157, 182525, 527509, 1524529, 4405969, 12733489, 36800465, 106355317, 307372573, 888323221, 2567301757, 7419639785, 21443156953, 61971873769, 179102039257, 517614500173, 1495933669445, 4323328543981

Offset

0,1

Comments

Number of tilings of the 0-mod-4 pillow of order n is a perfect square times a(n). [Propp, 1999, p. 271]

References

J. Propp, Enumeration of matchings: problems and progress, pp. 255-291 in L. J. Billera et al., eds, New Perspectives in Algebraic Combinatorics, Cambridge, 1999 (see Problem 12).

Links

A.H.M. Smeets, Table of n, a(n) for n = 0..2169

J. Propp, Updated article

J. Propp, Enumeration of matchings: problems and progress, in L. J. Billera et al. (eds.), New Perspectives in Algebraic Combinatorics

Index entries for linear recurrences with constant coefficients, signature (2,2,2,-1).

Formula

G.f.: (5 + 3*x + x^2 -x^3) / (1 - 2*x - 2*x^2 - 2*x^3 + x^4).

a(-n) = a(-5 + n). a(-1) = a(-2) = 1. a(n) = 2*a(n-1) + 2*a(n-2) + 2*a(n-3) - a(n-4). - Michael Somos, Dec 15 2011

A112835(2*n + 2) = a(n).

Lim_{n -> inf} a(n)/a(n-1) = A318605. - A.H.M. Smeets, Sep 12 2018

Example

G.f. = 5 + 13*x + 37*x^2 + 109*x^3 + 313*x^4 + 905*x^5 + 2617*x^6 + 7561*x^7 + ...

Maple

seq(coeff(series((5+3*x+x^2-x^3)/(1-2*x-2*x^2-2*x^3+x^4), x, n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Sep 12 2018

Mathematica

CoefficientList[Series[(5 + 3*x + x^2 -x^3)/(1 - 2*x - 2*x^2 - 2*x^3 + x^4), {x, 0, 50}], x] (* Stefano Spezia, Sep 12 2018 *)
LinearRecurrence[{2, 2, 2, -1}, {5, 13, 37, 109}, 30] (* Harvey P. Dale, Sep 03 2021 *)

Prog

(PARI) {a(n) = my(m = n+2); if( m < 0, m = -1 - m); polcoeff( (1 - x + x^2 - x^3) / (1 - 2*x - 2*x^2 - 2*x^3 + x^4) + x * O(x^m), m)}; /* Michael Somos, Dec 15 2011 */
(PARI) x='x+O('x^33); Vec((5+3*x+x^2-x^3)/(1-2*x-2*x^2-2*x^3+x^4)) \\ Altug Alkan, Sep 12 2018
(GAP) a:=[5, 13, 37, 109];; for n in [5..30] do a[n]:=2*a[n-1]+2*a[n-2]+2*a[n-3]-a[n-4]; od; a; # Muniru A Asiru, Sep 12 2018

Crossrefs

Cf. A112835.

Sequence in context: A193642 A220709 A182312 * A199108 A125734 A146925

Adjacent sequences: A071097 A071098 A071099 * A071101 A071102 A071103

Keyword

nonn,easy

Author

N. J. A. Sloane, May 28 2002

Status

approved