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
Examples
G.f. = 5 + 13*x + 37*x^2 + 109*x^3 + 313*x^4 + 905*x^5 + 2617*x^6 + 7561*x^7 + ...
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).
Crossrefs
Cf. A112835.
Programs
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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
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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
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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 *) -
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
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
Comments