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URL: https://oeis.org/A071101

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A071101
Expansion of (5 + 6*x + 3*x^2 - 2*x^3) / (1 - 2*x - 2*x^2 - 2*x^3 + x^4) in powers of x.
8
5, 16, 45, 130, 377, 1088, 3145, 9090, 26269, 75920, 219413, 634114, 1832625, 5296384, 15306833, 44237570, 127848949, 369490320, 1067846845, 3086134658, 8919094697, 25776662080, 74495936025, 215297250946, 622220603405, 1798250918672, 5197041610021
OFFSET
0,1
COMMENTS
Number of tilings of the 2-mod-4 pillow of order n is a perfect square times a(n). [Propp, 1999, p. 272]
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
J. Propp, Updated article
J. Propp, Enumeration of matchings: problems and progress, in L. J. Billera et al. (eds.), New Perspectives in Algebraic Combinatorics
FORMULA
G.f.: (5 + 6*x + 3*x^2 - 2*x^3) / (1 - 2*x - 2*x^2 - 2*x^3 + x^4).
a(-n) = a(-6 + n). a(-1) = 2, a(-2) = 1, a(-3) = 0. 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 + 3) = a(n).
Lim_{n -> inf} a(n)/a(n-1) = A318605. - A.H.M. Smeets, Sep 12 2018
EXAMPLE
G.f. = 5 + 16*x + 45*x^2 + 130*x^3 + 377*x^4 + 1088*x^5 + 3145*x^6 + 9090*x^7 + ...
MAPLE
seq(coeff(series((5+6*x+3*x^2-2*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
Table[Abs[Fibonacci[n+3, 1+I]]^2, {n, 0, 30}] (* Vladimir Reshetnikov, Oct 05 2016 *)
CoefficientList[Series[(5+6*x+3*x^2-2*x^3)/(1-2*x-2*x^2-2*x^3+x^4), {x, 0, 30}], x] (* Stefano Spezia, Sep 12 2018 *)
LinearRecurrence[{2, 2, 2, -1}, {5, 16, 45, 130}, 30] (* Harvey P. Dale, Oct 03 2024 *)
PROG
(PARI) {a(n) = my(m = abs(n+3)); polcoeff( (x - 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+6*x+3*x^2-2*x^3)/(1-2*x-2*x^2-2*x^3+x^4)) \\ Altug Alkan, Sep 12 2018
(Python)
from math import log
a0, a1, a2, a3, n = 130, 45, 16, 5, 3
print(0, a3)
print(1, a2)
print(2, a1)
print(3, a0)
while log(a0)/log(10) < 1000:
a0, a1, a2, a3, n = 2*(a0+a1+a2)-a3, a0, a1, a2, n+1
print(n, a0) # A.H.M. Smeets, Sep 12 2018
(GAP) a:=[5, 16, 45, 130];; 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
(Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (5+6*x+3*x^2-2*x^3)/(1-2*x-2*x^2-2*x^3+x^4) )); // G. C. Greubel, Jul 29 2019
(SageMath) ((5+6*x+3*x^2-2*x^3)/(1-2*x-2*x^2-2*x^3+x^4)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jul 29 2019
CROSSREFS
Sequence in context: A004146 A369803 A275126 * A110580 A351283 A055552
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, May 28 2002
STATUS
approved