A112835 Small-number statistic from the enumeration of domino tilings of a 3-pillow of order n.

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

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

Table of n, a(n) for n=0..34.

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.

Cf. A071100, A071101.

Sequence in context: A239340 A124201 A100953 * A206625 A176168 A308770

Adjacent sequences: A112832 A112833 A112834 * A112836 A112837 A112838

Keyword

easy,nonn

Author

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

Status

approved