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A112835 Small-number statistic from the enumeration of domino tilings of a 3-pillow of order n. 15
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; text; internal format)
OFFSET
0,2
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.
Plotting A112835(n+2)/A112835(n) gives an intriguing damped sine curve.
REFERENCES
C. Hanusa (2005). A Gessel-Viennot-Type Method for Cycle Systems with Applications to Aztec Pillows. PhD Thesis. University of Washington, Seattle, USA.
LINKS
A D Mednykh, I A Mednykh, The number of spanning trees in circulant graphs, its arithmetic properties and asymptotic, arXiv preprint arXiv:1711.00175, 2017. See Section 4.
FORMULA
a(2*n + 2) = A071100(n). a(2*n + 3) = A071101(n).
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
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.
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 */
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.
Sequence in context: A239340 A124201 A100953 * A206625 A176168 A308770
KEYWORD
easy,nonn
AUTHOR
Christopher Hanusa (chanusa(AT)math.binghamton.edu), Sep 21 2005
STATUS
approved

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Last modified June 19 11:49 EDT 2024. Contains 373503 sequences. (Running on oeis4.)