A332862 -id:A332862 - cp's OEIS frontend

A054894 a(n+1) = 4a(n) + 4a(n-1) - 4*a(n-2) - a(n-3) with a(1)=1, a(2)=2, a(3)=11, a(4)=48.

1, 2, 11, 48, 227, 1054, 4921, 22944, 107017, 499106, 2327795, 10856592, 50634107, 236152510, 1101392305, 5136786240, 23957470033, 111735303362, 521122556315, 2430464772336, 11335450631123, 52867436085214, 246568565219689, 1149971737922784, 5363356017597913, 25014169325118818

Offset: 1

Barry Cipra, Jul 04 2000

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Svenja Huntemann, Neil A. McKay, Counting Domineering Positions, arXiv:1909.12419 [math.CO], 2019.
Index entries for linear recurrences with constant coefficients, signature (4,4,-4,-1).

Column 3 of A332862 (apart from the initial term 1 here).

Cf. A055518, A055519.

a:=[1,2,11,48];; for n in [5..40] do a[n]:=4*(a[n-1]+a[n-2]-a[n-3]) -a[n-4]; od; a; # G. C. Greubel, Dec 29 2019

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( x*(1-2*x-x^2)/(1-4*x-4*x^2+4*x^3+x^4) )); // G. C. Greubel, Dec 29 2019

seq(coeff(series(x^4/((1+2*x)*(2*x^3+x^2-2*x+1)), x, n+1), x, n), n = 0..40); # G. C. Greubel, Dec 29 2019

CoefficientList[Series[(1-2*x-x^2)/(1-4*x-4*x^2+4*x^3+x^4), {x, 0, 40}], x] (* Vincenzo Librandi, May 05 2013 *)
LinearRecurrence[{4,4,-4,-1},{1,2,11,48},40] (* Harvey P. Dale, May 28 2014 *)

my(x='x+O('x^66)); Vec(x*(1-2*x-x^2)/(1-4*x-4*x^2+4*x^3+x^4)) \\ Joerg Arndt, May 06 2013

def A054894_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(1-2*x-x^2)/(1-4*x-4*x^2+4*x^3+x^4) ).list()
a=A054894_list(40); a[1:] # G. C. Greubel, Dec 29 2019

a(n) = Sum_{k=1..n} Fibonacci(k)^3*a(n-k), with a(0)=1. - Vladeta Jovovic, Apr 23 2003

G.f.: x*(1-2*x-x^2)/(1-4*x-4*x^2+4*x^3+x^4). - Vaclav Kotesovec, Nov 27 2012

A329707 Number of placements of zero or more dominoes on a 2 X n grid where no two empty squares are horizontally adjacent.

Original entry on oeis.org

1, 2, 4, 11, 25, 61, 146, 351, 844, 2028, 4875, 11717, 28163, 67692, 162703, 391070, 939968, 2259289, 5430383, 13052363, 31372406, 75406105, 181244648, 435636112, 1047086489, 2516756727, 6049227537, 14539805696, 34947594281, 83999358146, 201899224084, 485281049587, 1166412095721

Offset: 0

Andrew Howroyd, Feb 28 2020

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,1,0,0,0,-1).

Row 2 of A332862.

LinearRecurrence[{2, 1, 0, 0, 0, -1}, {1, 2, 4, 11, 25, 61}, 50] (* Paolo Xausa, Jun 19 2024 *)

Vec((1 - x)*(1 + x + x^3)/(1 - 2*x - x^2 + x^6) + O(x^30))

a(n) = 2*a(n-1) + a(n-2) - a(n-6) for n > 6.

G.f.: (1 - x)*(1 + x + x^3)/(1 - 2*x - x^2 + x^6).

A332865 Number of placements of zero or more dominoes on the n X n grid where no two empty squares are horizontally adjacent.

Original entry on oeis.org

1, 4, 48, 1427, 140555, 40008789, 33656587715, 84588476099284, 626461671945179295, 13776144517953719025396, 897220763259635483826935324, 173109540246969825014223808529273, 98978509126162805673620043358494745638, 167661422725328648892707605323564506782035252

Offset: 1

Neil A. McKay, Feb 27 2020

nonn

The number of positions of n X n Domineering where horizontal (Right) has no moves, also called Right ends. Domineering is a game in which players take turns placing dominoes on a grid, one player placing vertically and the other horizontally until the player to place cannot place a domino.

Bjorn Huntemann, Svenja Huntemann, and Neil A. McKay SageMath code for Counting Domineering Positions
Svenja Huntemann and Neil A. McKay, Counting Domineering Positions, arXiv:1909.12419 [math.CO], 2019.

Main diagonal of A332862.
Cf. A287595 (the number of placements of dominoes on an n X n grid where no two empty squares are horizontally or vertically adjacent).
Cf. A332714.

# See Bjorn Huntemann, Svenja Huntemann, Neil A. McKay link.

a(9)-a(14) from Andrew Howroyd, Feb 28 2020

cp's OEIS Frontend

A054894 a(n+1) = 4a(n) + 4a(n-1) - 4*a(n-2) - a(n-3) with a(1)=1, a(2)=2, a(3)=11, a(4)=48.

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A329707 Number of placements of zero or more dominoes on a 2 X n grid where no two empty squares are horizontally adjacent.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A332865 Number of placements of zero or more dominoes on the n X n grid where no two empty squares are horizontally adjacent.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Sage

Extensions

cp's OEIS Frontend

A054894 a(n+1) = 4*a(n) + 4*a(n-1) - 4*a(n-2) - a(n-3) with a(1)=1, a(2)=2, a(3)=11, a(4)=48.

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A329707 Number of placements of zero or more dominoes on a 2 X n grid where no two empty squares are horizontally adjacent.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A332865 Number of placements of zero or more dominoes on the n X n grid where no two empty squares are horizontally adjacent.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Sage

Extensions

A054894 a(n+1) = 4a(n) + 4a(n-1) - 4*a(n-2) - a(n-3) with a(1)=1, a(2)=2, a(3)=11, a(4)=48.