A194540 -id:A194540 - cp's OEIS frontend

A055244 Number of certain stackings of n+1 squares on a double staircase.

1, 1, 3, 6, 12, 23, 43, 79, 143, 256, 454, 799, 1397, 2429, 4203, 7242, 12432, 21271, 36287, 61739, 104791, 177476, 299978, 506111, 852457, 1433593, 2407443, 4037454, 6762708, 11314391, 18909139, 31569799, 52657247, 87751624

Offset: 0

Wolfdieter Lang, May 10 2000

a(n)= G_{n+1} of Turban reference eq.(3.9).
Equals A046854 * [1,2,3,...]. - Gary W. Adamson, Dec 23 2008
(1 + x + 3x^2 + 6x^3 + ...) = (1 + x + 2x^2 + 3x^3 + 5x^4 + 8x^5 + ...) * (1 + x^2 + 2x^3 + 3x^4 + 5x^5 + 8x^6 + ...). -Gary W. Adamson, Jul 27 2010
Column 1 of A194540. - R. H. Hardin, Aug 28 2011

L. Turban, Lattice animals on a staircase and Fibonacci numbers, J.Phys. A 33 (2000) 2587-2595.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Arnold Adelberg and Tamás Lengyel, New results on the 2-adic valuation of the central Stirling numbers S(2k, k), Harvey Mudd College (2024).
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

Cf. A000045, A001629, A046854, A055245.

a:= n-> (Matrix([[1,-1,2,-4]]). Matrix(4, (i,j)-> if (i=j-1) then 1 elif j=1 then [2,1,-2,-1][i] else 0 fi)^(n))[1,1] ; seq (a(n), n=0..33); # Alois P. Heinz, Aug 05 2008

a[0] = a[1] = 1; a[n_] := a[n] = (((n-4)*n-6)*a[n-2] + ((n-5)*n-11)*a[n-1]) / ((n-6)*n-1); Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 11 2014 *)
CoefficientList[Series[(1 - x + x^3)/(1 - x - x^2)^2, {x, 0, 50}], x] (* Vincenzo Librandi, Mar 13 2014 *)
LinearRecurrence[{2,1,-2,-1},{1,1,3,6},60] (* Harvey P. Dale, Jul 13 2022 *)

G.f.: (1-x+x^3)/(1-x-x^2)^2. (from Turban reference eq.(3.3) with t=1).
a(n) = ((n+5)*F(n+1)+(2*n-3)*F(n))/5 with F(n)=A000045(n) (Fibonacci numbers) (from Turban reference eq.(3.9)).
a(n) = A001629(n+1) + F(n-1). - Gary W. Adamson, Jul 27 2007
a(n) = (((n-4)*n-6)*a(n-2) + ((n-5)*n-11)*a(n-1)) / ((n-6)*n-1). - Jean-François Alcover, Mar 11 2014

A194535 Number of lower triangles of an n X n 0..3 array with new values introduced in row major order 0..3 and no element equal to more than one horizontal or vertical neighbor.

Original entry on oeis.org

1, 4, 116, 16556, 7721920, 11525456507, 54979732214116, 838387866908847478, 40866499579695484000499, 6367568042830726931523028453, 3171485572844765112503023656979959

Offset: 1

R. H. Hardin Aug 28 2011

nonn

Column 3 of A194540

			Some solutions for 3X3
..0......0......0......0......0......0......0......0......0......0......0
..1.2....1.1....1.2....1.2....1.1....1.2....1.2....1.2....1.0....1.2....1.1
..3.3.0..0.0.2..2.3.1..0.3.0..2.0.1..3.2.1..0.3.1..2.1.3..2.1.2..2.0.1..2.2.1

R. H. Hardin, Table of n, a(n) for n = 1..12

A194534 Number of lower triangles of an n X n 0..2 array with new values introduced in row major order 0..2 and no element equal to more than one horizontal or vertical neighbor.

Original entry on oeis.org

1, 4, 59, 1851, 119548, 16039294, 4460214471, 2572187445993, 3075932897564028, 7627704068111688574, 39223874584658826539499, 418261269510360187893175534, 9248796169262343437995449320699, 424095060368117191810477243815045914

Offset: 1

R. H. Hardin Aug 28 2011

nonn

Column 2 of A194540

			Some solutions for 3X3
..0......0......0......0......0......0......0......0......0......0......0
..0.1....0.1....1.2....1.1....1.2....1.1....1.1....1.2....1.1....1.0....1.1
..2.0.0..1.0.0..0.2.0..2.0.0..2.1.2..0.2.1..2.2.1..0.1.0..2.0.2..1.2.0..0.2.0

A194536 Number of lower triangles of an n X n 0..4 array with new values introduced in row major order 0..4 and no element equal to more than one horizontal or vertical neighbor.

Original entry on oeis.org

1, 4, 131, 43785, 79201795, 627974590650, 21399815343394998, 3132305932642809585983, 1969210116072259125767299138

Offset: 1

R. H. Hardin, Aug 28 2011

nonn

Column 4 of A194540.

			Some solutions for 3 X 3
..0......0......0......0......0......0......0......0......0......0......0
..1.2....1.0....1.2....1.1....1.0....1.2....1.0....0.1....1.2....1.2....0.1
..2.0.1..2.2.1..0.1.0..0.2.1..2.3.4..3.2.0..0.2.3..1.0.0..2.1.2..0.3.1..1.0.1

Cf. A194540.

A194537 Number of lower triangles of an n X n 0..5 array with new values introduced in row major order 0..5 and no element equal to more than one horizontal or vertical neighbor.

Original entry on oeis.org

1, 4, 132, 62038, 286773762, 8096809998526, 1243336605965267360, 1026437889413641002431445

Offset: 1

R. H. Hardin Aug 28 2011

nonn

Column 5 of A194540

			Some solutions for 3X3
..0......0......0......0......0......0......0......0......0......0......0
..1.2....1.2....1.2....1.2....1.2....1.2....0.1....1.0....1.1....0.1....0.1
..2.0.3..0.2.1..3.1.1..3.2.1..3.0.1..3.4.1..1.0.1..2.0.1..0.0.2..1.0.2..1.0.0

A194538 Number of lower triangles of an n X n 0..6 array with new values introduced in row major order 0..6 and no element equal to more than one horizontal or vertical neighbor.

Original entry on oeis.org

1, 4, 132, 67377, 543350470, 40994098419294, 21253155706215768591, 71381360690596538575641162

Offset: 1

R. H. Hardin Aug 28 2011

nonn

Column 6 of A194540

			Some solutions for 3X3
..0......0......0......0......0......0......0......0......0......0......0
..0.1....1.2....1.2....1.1....1.2....1.2....1.1....1.0....0.1....1.2....1.1
..2.2.0..1.0.2..0.3.3..2.0.0..2.3.4..2.0.0..2.2.3..2.1.3..2.3.0..2.3.1..2.0.1

A194539 Number of lower triangles of an n X n 0..7 array with new values introduced in row major order 0..7 and no element equal to more than one horizontal or vertical neighbor.

Original entry on oeis.org

1, 4, 132, 68106, 703256449, 108504879480483, 153378412416345530447

Offset: 1

R. H. Hardin Aug 28 2011

nonn

Column 7 of A194540

			Some solutions for 3X3
..0......0......0......0......0......0......0......0......0......0......0
..1.1....1.1....1.0....1.2....0.1....1.2....1.2....1.2....0.1....1.2....1.2
..2.2.0..0.2.0..0.2.3..1.3.3..2.3.1..0.3.4..1.3.2..2.0.3..2.2.1..0.3.2..0.1.2

cp's OEIS Frontend

A055244 Number of certain stackings of n+1 squares on a double staircase.

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A194535 Number of lower triangles of an n X n 0..3 array with new values introduced in row major order 0..3 and no element equal to more than one horizontal or vertical neighbor.

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A194534 Number of lower triangles of an n X n 0..2 array with new values introduced in row major order 0..2 and no element equal to more than one horizontal or vertical neighbor.

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A194536 Number of lower triangles of an n X n 0..4 array with new values introduced in row major order 0..4 and no element equal to more than one horizontal or vertical neighbor.

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A194537 Number of lower triangles of an n X n 0..5 array with new values introduced in row major order 0..5 and no element equal to more than one horizontal or vertical neighbor.

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A194538 Number of lower triangles of an n X n 0..6 array with new values introduced in row major order 0..6 and no element equal to more than one horizontal or vertical neighbor.

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A194539 Number of lower triangles of an n X n 0..7 array with new values introduced in row major order 0..7 and no element equal to more than one horizontal or vertical neighbor.

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