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A112835
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Small-number statistic from the enumeration of domino tilings of a 3-pillow of order n.
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15
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1, 2, 5, 5, 13, 16, 37, 45, 109, 130, 313, 377, 905, 1088, 2617, 3145, 7561, 9090, 21853, 26269, 63157, 75920, 182525, 219413, 527509, 634114, 1524529, 1832625, 4405969, 5296384, 12733489, 15306833, 36800465, 44237570, 106355317
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| 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.
Breaking this sequence into two sequences yields A071101 and A071100, both of which have nice generating functions.
Plotting A112835(n+2)/A112835(n) gives an intriguing damped sine curve.
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REFERENCES
| C. Hanusa (2005). A Gessel-Viennot-Type Method for Cycle Systems with Applications to Aztec Pillows. PhD Thesis. University of Washington, Seattle, USA.
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FORMULA
| 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
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
a(2*n + 2) = A071100(n). a(2*n + 3) = A071101(n). - Michael Somos, Dec 15 2011
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EXAMPLE
| 1 + 2*x + 5*x^2 + 5*x^3 + 13*x^4 + 16*x^5 + 37*x^6 + 45*x^7 + 109*x^8 + ...
The number of domino tilings of the 3-pillow of order 4 is 117=3^2*13. A112835(4)=13.
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PROG
| (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 */
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CROSSREFS
| A112833 breaks down as A112834^2 times A112835, where A112835 is not necessarily squarefree.
5-pillows: A112836-A112838; 7-pillows: A112839-A112841; 9-pillows: A112842-A112844.
Cf. A071100, A071101.
Sequence in context: A124201 A100953 * A206625 A176168 A154692 A144293
Adjacent sequences: A112832 A112833 A112834 * A112836 A112837 A112838
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KEYWORD
| easy,nonn
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AUTHOR
| Christopher Hanusa (chanusa(AT)math.binghamton.edu), Sep 21 2005
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