cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-14 of 14 results.

A112840 Large-number statistic from the enumeration of domino tilings of a 7-pillow of order n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 7, 11, 28, 51, 154, 389, 1556, 4833, 22477, 80532, 440512, 1916580, 13388593, 73763989, 632754664, 4175659899, 42606281476, 336819337955, 4181786155008, 40981322633555, 630857431556758, 7576627032674784
Offset: 0

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Author

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

Keywords

Comments

A 7-pillow is a generalized Aztec pillow. The 7-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 7 horizontal squares to every 1 vertical square and steps down with steps of 1 horizontal square to every 1 vertical square.

Examples

			The number of domino tilings of the 7-pillow of order 8 is 23353=11^2*193. A112840(n)=11.

References

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

Crossrefs

A112839 breaks down as A112840^2 times A112841, where A112841 is not necessarily squarefree.
3-pillows: A112833-A112835; 5-pillows: A112836-A112838; 9-pillows: A112842-A112844.

A112843 Large-number statistic from the enumeration of domino tilings of a 9-pillow of order n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 3, 7, 11, 26, 44, 118, 221, 677, 1721, 6884, 21165, 95800, 324693, 1633462, 6253408, 35917622, 161554715, 1151376732, 6387653627, 54325024024, 348582834189, 3376194023305, 24664208882500, 273518249356480
Offset: 0

Views

Author

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

Keywords

Comments

A 9-pillow is a generalized Aztec pillow. The 9-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 9 horizontal squares to every 1 vertical square and steps down with steps of 1 horizontal square to every 1 vertical square.

Examples

			The number of domino tilings of the 9-pillow of order 8 is 9065=7^2*185. A112843(n)=7.

References

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

Crossrefs

A112842 breaks down as A112843^2 times A112844, where A112844 is not necessarily squarefree.
3-pillows: A112833-A112835; 5-pillows: A112836-A112838; 7-pillows: A112839-A112841.

A112837 Large-number statistic from the enumeration of domino tilings of a 5-pillow of order n.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 7, 12, 35, 87, 348, 1107, 5518, 22464, 150574, 817057, 7118856, 49644383, 560434040, 5142118400, 76370120248, 914476059335, 17638655014128, 274908897964359, 6936239946318204, 141510942505315328
Offset: 0

Views

Author

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

Keywords

Comments

A 5-pillow is a generalized Aztec pillow. The 5-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 5 horizontal squares to every 1 vertical square and steps down with steps of 1 horizontal square to every 1 vertical square.

Examples

			The number of domino tilings of the 5-pillow of order 6 is 1666=7^2*34. A112837(n)=7.

References

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

Crossrefs

A112833 breaks down as A112834^2 times A112835, where A112835 is not necessarily squarefree.
3-pillows: A112833-A112835; 7-pillows: A112839-A112841; 9-pillows: A112842-A112844.

A206625 Expansion of x * (1 + x) * (1 - x^2) * (1 + x^3) / (1 - 2*x^2 - 2*x^4 - 2*x^6 + x^8) in powers of x.

Original entry on oeis.org

0, 1, 1, 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

Views

Author

Michael Somos, Feb 10 2012

Keywords

Comments

This is a divisibility sequence; that is, if n divides m, then a(n) divides a(m).

Examples

			G.f. = x + x^2 + x^3 + 2*x^4 + 5*x^5 + 5*x^6 + 13*x^7 + 16*x^8 + 37*x^9 + ...

References

  • J. A. Sjogren, Cycles and spanning trees. Math. Comput. Modelling 15, No.9, 87-102 (1991).

Links

Crossrefs

Cf. A071100 (bisection), A071101 (bisection), A112835, A138573.

Programs

  • Magma
    m:=25; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(1+x)*(1-x^2)*(1+x^3)/(1-2*x^2-2*x^4-2*x^6+x^8 ))); // G. C. Greubel, Aug 12 2018
  • Mathematica
    CoefficientList[Series[x*(1+x)*(1-x^2)*(1+x^3)/(1-2*x^2-2*x^4-2*x^6+x^8 ), {x, 0, 50}], x] (* G. C. Greubel, Aug 12 2018 *)
  • PARI
    {a(n) = my(m = abs(n)); polcoeff( x * (1 + x) * (1 - x^2) * (1 + x^3) / (1 - 2*x^2 - 2*x^4 - 2*x^6 + x^8) + x * O(x^m), m)};
  • PARI
    {a(n) = my(m = abs(n), v); v = polroots( Pol([ 1, 2, 4, 2, 1])); sqrtint( round( prod( k=1, 4, v[k]^m - 1, 2^(m%2) / 20)))};

Formula

G.f.: x * (1 + x) * (1 - x^2) * (1 + x^3) / (1 - 2*x^2 - 2*x^4 - 2*x^6 + x^8).
a(n) = a(-n) = 2*a(n-2) + 2*a(n-4) + 2*a(n-6) - a(n-8) for all n in Z.
a(2*n + 5) = A071100(n). a(2*n + 6) = A071101(n). a(n + 3) = A112835(n). a(2*n) = A138573(n).
Previous Showing 11-14 of 14 results.

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