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Small-number statistic from the enumeration of domino tilings of a 3-pillow of order n.
15

%I #14 Jan 20 2018 11:04:22

%S 1,2,5,5,13,16,37,45,109,130,313,377,905,1088,2617,3145,7561,9090,

%T 21853,26269,63157,75920,182525,219413,527509,634114,1524529,1832625,

%U 4405969,5296384,12733489,15306833,36800465,44237570,106355317

%N Small-number statistic from the enumeration of domino tilings of a 3-pillow of order n.

%C A 3-pillow is also called an Aztec pillow. The 3-pillow of order n is a rotationally-symmetric region. It has a 2 X 2n central band of squares and then steps up from this band with steps of 3 horizontal squares to every 1 vertical square and steps down with steps of 1 horizontal square to every 1 vertical square.

%C Plotting A112835(n+2)/A112835(n) gives an intriguing damped sine curve.

%D C. Hanusa (2005). A Gessel-Viennot-Type Method for Cycle Systems with Applications to Aztec Pillows. PhD Thesis. University of Washington, Seattle, USA.

%H A D Mednykh, I A Mednykh, <a href="http://arxiv.org/abs/1711.00175">The number of spanning trees in circulant graphs, its arithmetic properties and asymptotic</a>, arXiv preprint arXiv:1711.00175, 2017. See Section 4.

%F a(2*n + 2) = A071100(n). a(2*n + 3) = A071101(n).

%F G.f.: (1 + x - x^2 + x^4 - x^5 - x^6) / (1 - 2*x^2 - 2*x^4 - 2*x^6 + x^8) = (1 + x) * (1 - x^2) * (1 + x^3) / (1 - 2*x^2 - 2*x^4 - 2*x^6 + x^8). - _Michael Somos_, Dec 15 2011

%F a(-n) = a(-6 + n). a(-1) = a(-2) = 1, a(-3) = 0. a(n) = 2*a(n-2) + 2*a(n-4) + 2*a(n-6) - a(n-8). - _Michael Somos_, Dec 15 2011

%e 1 + 2*x + 5*x^2 + 5*x^3 + 13*x^4 + 16*x^5 + 37*x^6 + 45*x^7 + 109*x^8 + ...

%e The number of domino tilings of the 3-pillow of order 4 is 117=3^2*13. A112835(4)=13.

%o (PARI) {a(n) = local(m = abs(n+3)); polcoeff( (x + x^2 - x^3 + x^5 - x^6 - x^7) / (1 - 2*x^2 - 2*x^4 - 2*x^6 + x^8) + x * O(x^m), m)} /* _Michael Somos_, Dec 15 2011 */

%Y A112833 breaks down as A112834^2 times A112835, where A112835 is not necessarily squarefree.

%Y 5-pillows: A112836-A112838; 7-pillows: A112839-A112841; 9-pillows: A112842-A112844.

%Y Cf. A071100, A071101.

%K easy,nonn

%O 0,2

%A Christopher Hanusa (chanusa(AT)math.binghamton.edu), Sep 21 2005