A095939 -id:A095939 - cp's OEIS frontend

A228754 T(n,k)=Number of nXk binary arrays with top left element equal to 1 and no two ones adjacent horizontally or antidiagonally.

1, 1, 2, 2, 3, 4, 3, 9, 8, 8, 5, 20, 39, 21, 16, 8, 50, 126, 168, 55, 32, 13, 119, 482, 780, 723, 144, 64, 21, 289, 1712, 4599, 4808, 3111, 377, 128, 34, 696, 6277, 24246, 43862, 29608, 13386, 987, 256, 55, 1682, 22700, 134440, 342207, 418370, 182288, 57597, 2584, 512

Offset: 1

R. H. Hardin Sep 02 2013

Table starts
...1....1......2.......3.........5...........8...........13.............21
...2....3......9......20........50.........119..........289............696
...4....8.....39.....126.......482........1712.........6277..........22700
...8...21....168.....780......4599.......24246.......134440.........728537
..16...55....723....4808.....43862......342207......2876170.......23326164
..32..144...3111...29608....418370.....4823826.....61534448......746135864
..64..377..13386..182288...3990739....67970044...1316714732....23857469157
.128..987..57597.1122240..38067290...957616341..28177227352...762713002760
.256.2584.247827.6908896.363121586.13491214832.602998827928.24382157716612

			Some solutions for n=4 k=4
..1..0..1..0....1..0..0..1....1..0..0..0....1..0..0..1....1..0..1..0
..0..0..0..1....1..0..0..0....1..0..0..0....0..1..0..0....1..0..0..1
..0..0..0..0....1..0..0..1....0..0..0..0....0..0..0..1....0..1..0..1
..0..0..0..1....0..0..0..0....0..1..0..0....0..1..0..0....0..0..0..0

R. H. Hardin, Table of n, a(n) for n = 1..1347

Column 1 is A000079(n-1)
Column 2 is A001906
Column 3 is A095939
Row 1 is A000045
Row 2 is A097075(n+1)

Empirical for column k:
k=1: a(n) = 2*a(n-1)
k=2: a(n) = 3*a(n-1) -a(n-2)
k=3: a(n) = 5*a(n-1) -3*a(n-2)
k=4: a(n) = 8*a(n-1) -12*a(n-2) +4*a(n-3)
k=5: a(n) = 13*a(n-1) -36*a(n-2) +29*a(n-3) -5*a(n-4) for n>5
k=6: a(n) = 21*a(n-1) -112*a(n-2) +217*a(n-3) -157*a(n-4) +36*a(n-5) for n>6
k=7: [order 7] for n>9
Empirical for row n:
n=1: a(n) = a(n-1) +a(n-2)
n=2: a(n) = a(n-1) +3*a(n-2) +a(n-3)
n=3: a(n) = 2*a(n-1) +6*a(n-2) -a(n-4)
n=4: [order 8]
n=5: [order 13]
n=6: [order 21]
n=7: [order 34]

A095934 Expansion of (1-x)^2/(1-5x+3x^2).

Original entry on oeis.org

1, 3, 13, 56, 241, 1037, 4462, 19199, 82609, 355448, 1529413, 6580721, 28315366, 121834667, 524227237, 2255632184, 9705479209, 41760499493, 179686059838, 773148800711, 3326685824041, 14313982718072, 61589856118237, 265007332436969, 1140267093830134

Offset: 0

N. J. A. Sloane, Jul 13 2004

a(n) is the number of generalized compositions of n when there are i+2 different types of i, (i=1,2,...). [Milan Janjic, Sep 24 2010]

P. J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 89-102.
P. J. Cameron, Some sequences of integers, in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.
Index entries for linear recurrences with constant coefficients, signature (5,-3).

Cf. A003480, A010903, A010920; equals A095939 + A095940.

CoefficientList[Series[(1-x)^2/(1-5x+3x^2),{x,0,30}],x] (* or *) LinearRecurrence[{5,-3},{1,3,13},30] (* Harvey P. Dale, Jun 21 2021 *)

a(n)=polcoeff((1-x)^2/(1-5*x+3*x^2)+x*O(x^n),n)

a(n+2) = 5a(n+1) - 3a(n) (n >= 1); a(0) = 1, a(1) = 3, a(2) = 13.

A095940 a(n+2) = 5a(n+1) - 3a(n) (n >= 1); a(0) = 0, a(1) = 1, a(2) = 4.

Original entry on oeis.org

0, 1, 4, 17, 73, 314, 1351, 5813, 25012, 107621, 463069, 1992482, 8573203, 36888569, 158723236, 682950473, 2938582657, 12644061866, 54404561359, 234090621197, 1007239421908, 4333925245949, 18647907964021, 80237764082258

Offset: 0

N. J. A. Sloane, Jul 13 2004

Index entries for linear recurrences with constant coefficients, signature (5,-3).

Cf. A018902; equals A095934 - A095939.

Join[{0},LinearRecurrence[{5,-3},{1,4},30]] (* Harvey P. Dale, Aug 20 2011 *)

G.f.: (x-x^2)/(3*x^2-5*x+1). - Harvey P. Dale, Aug 20 2011

Extended by Ray Chandler, Jul 16 2004

cp's OEIS Frontend

A228754 T(n,k)=Number of nXk binary arrays with top left element equal to 1 and no two ones adjacent horizontally or antidiagonally.

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A095934 Expansion of (1-x)^2/(1-5x+3x^2).

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PARI

Formula

A095940 a(n+2) = 5a(n+1) - 3a(n) (n >= 1); a(0) = 0, a(1) = 1, a(2) = 4.

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cp's OEIS Frontend

A228754 T(n,k)=Number of nXk binary arrays with top left element equal to 1 and no two ones adjacent horizontally or antidiagonally.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A095934 Expansion of (1-x)^2/(1-5*x+3*x^2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A095940 a(n+2) = 5*a(n+1) - 3*a(n) (n >= 1); a(0) = 0, a(1) = 1, a(2) = 4.

Views

Author

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Programs

Mathematica

Formula

Extensions

A095934 Expansion of (1-x)^2/(1-5x+3x^2).

A095940 a(n+2) = 5a(n+1) - 3a(n) (n >= 1); a(0) = 0, a(1) = 1, a(2) = 4.