A077909 -id:A077909 - cp's OEIS frontend

A128099 Triangle read by rows: T(n,k) is the number of ways to tile a 3 X n rectangle with k pieces of 2 X 2 tiles and 3n-4k pieces of 1 X 1 tiles (0 <= k <= floor(n/2)).

1, 1, 1, 2, 1, 4, 1, 6, 4, 1, 8, 12, 1, 10, 24, 8, 1, 12, 40, 32, 1, 14, 60, 80, 16, 1, 16, 84, 160, 80, 1, 18, 112, 280, 240, 32, 1, 20, 144, 448, 560, 192, 1, 22, 180, 672, 1120, 672, 64, 1, 24, 220, 960, 2016, 1792, 448, 1, 26, 264, 1320, 3360, 4032, 1792, 128, 1, 28

Offset: 0

Emeric Deutsch, Feb 18 2007

Row sums are the Jacobsthal numbers (A001045).
Apparently, T(n,k)/2^n equals the probability P that n will occur as a partial sum in a randomly-generated infinite sequence of 1s and 2s with n compositions (ordered partitions) into (n-2k) 1s and k 2s. Example: T(6,2)=24; P = 3/8 (24/2^6) that 6 will occur as a partial sum in the sequence with 2 (6-2*2) 1s and 2 2s. - Bob Selcoe, Jul 06 2013
From Johannes W. Meijer, Aug 28 2013: (Start)
The antidiagonal sums are A077949 and the backwards antidiagonal sums are A052947.
Moving the terms in each column of this triangle, see the example, upwards to row 0 gives the Pell-Jacobsthal triangle A013609 as a square array. (End)
The numbers in rows of the triangle are along "first layer" skew diagonals pointing top-right in center-justified triangle given in A013609 ((1+2*x)^n) and along (first layer) skew diagonals pointing top-left in center-justified triangle given in A038207 ((2+x)^n), see links. - Zagros Lalo, Jul 31 2018
If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 2.000..., when n approaches infinity. - Zagros Lalo, Jul 31 2018
It appears that the rows of this array are the coefficients of the Jacobsthal polynomials (see MathWorld link). - Michel Marcus, Jun 15 2019

			Triangle starts:
  1;
  1;
  1,  2;
  1,  4;
  1,  6,  4;
  1,  8, 12;
  1, 10, 24,  8;
  1, 12, 40, 32;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 80-83, 357-358

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened
Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, and Graça Tomaz, Intrinsic Properties of a Non-Symmetric Number Triangle, J. Int. Seq., Vol. 26 (2023), Article 23.4.8.
Richard Fors, Independence Complexes of Certain Families of Graphs, Master's thesis in Mathematics at KTH, presented Aug 19 2011.
R. J. Mathar, Tiling n x m rectangles with 1 x 1 and s x s squares arXiv:1609.03964 [math.CO] (2016).
Zagros Lalo, First layer skew diagonals in center-justified triangle of coefficients in expansion of (1 + 2x)^n
Zagros Lalo, First layer skew diagonals in center-justified triangle of coefficients in expansion of (2 + x)^n
Eric Weisstein's World of Mathematics, Jacobsthal Polynomial

Cf. (Triangle sums) A001045, A095977, A077949, A052947, A113726, A052942, A077909.
Cf. (Similar triangles) A008315, A011973, A102541.
Cf. A013609, A038207
Cf. A207538

T := proc(n,k) if k<=n/2 then 2^k*binomial(n-k,k) else 0 fi end: for n from 0 to 16 do seq(T(n,k),k=0..floor(n/2)) od; # yields sequence in triangular form
T := proc(n, k) option remember: if k<0 or k > floor(n/2) then return(0) fi: if k = 0 then return(1) fi: 2*procname(n-2, k-1) + procname(n-1, k): end: seq(seq(T(n, k), k=0..floor(n/2)), n=0..13); # Johannes W. Meijer, Aug 28 2013

Table[2^k*Binomial[n - k, k] , {n,0,25}, {k,0,Floor[n/2]}] // Flatten  (* G. C. Greubel, Dec 28 2016 *)
t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, t[n - 1, k] + 2 t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 15}, {k, 0, Floor[n/2]}] // Flatten (* Zagros Lalo, Jul 31 2018 *)

T(n, k) = 2^k*binomial(n-k,k) = 2^k*A011973(n,k).
G.f.: 1/(1-z-2*t*z^2).
Sum_{k=0..floor(n/2)} k*T(n,k) = A095977(n-1).
From Johannes W. Meijer, Aug 28 2013: (Start)
T(n, k) = 2*T(n-2, k-1) + T(n-1, k) with T(n, 0) = 1 and T(n, k) = 0 for k < 0 and k > floor(n/2).
T(n, k) = A013609(n-k, k), n >= 0 and 0 <= k <= floor(n/2). (End)

A234312 Number of tilings of a 5 X n rectangle using n pentominoes of shapes L, X.

Original entry on oeis.org

1, 0, 2, 0, 4, 2, 8, 8, 16, 24, 36, 64, 88, 160, 224, 392, 576, 960, 1472, 2368, 3728, 5888, 9376, 14720, 23488, 36896, 58752, 92544, 146944, 232064, 367680, 581632, 920448, 1457152, 2305024, 3649664, 5773312, 9140224, 14460928, 22890496, 36221184, 57327616

Offset: 0

Alois P. Heinz, Dec 23 2013

nonn

			a(4) = 4:
._______.  ._______.  ._______.  ._______.
|_. |_. |  | ._| ._|  |_. | ._|  | ._|_. |
| | | | |  | | | | |  | | | | |  | | | | |
| | | | |  | | | | |  | | | | |  | | | | |
| |_| |_|  |_| |_| |  | |_|_| |  |_| | |_|
|___|___|  |___|___|  |___|___|  |___|___|.
a(5) = 2:
._________.  ._________.
| | ._____|  |_____. | |
| |_| |_. |  | ._| |_| |
| |_. ._| |  | |_. ._| |
|___|_| | |  | | |_|___|
|_______|_|  |_|_______|.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Pentomino
Index entries for linear recurrences with constant coefficients, signature (0,2,0,0,2)

Cf. A077909, A174249, A233427, A234931, A247125, A264812.

a:= n-> (<<0|1|0|0|0>, <0|0|1|0|0>, <0|0|0|1|0>,
          <0|0|0|0|1>, <2|0|0|2|0>>^n)[5, 5]:
seq(a(n), n=0..50);

LinearRecurrence[{0, 2, 0, 0, 2}, {1, 0, 2, 0, 4}, 50] (* Jean-François Alcover, May 28 2019 *)

G.f.: -1/(2*x^5+2*x^2-1).

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

A247126 Number of tilings of a 5 X n rectangle using n pentominoes of shapes F, U, X, N.

Original entry on oeis.org

1, 0, 0, 1, 2, 0, 1, 4, 4, 1, 14, 12, 17, 32, 64, 81, 138, 272, 489, 764, 1548, 2809, 5062, 9420, 17721, 32712, 60992, 114105, 213890, 398784, 747745, 1401476, 2624004, 4916369, 9218118, 17274340, 32378521, 60694768, 113785984, 213293721, 399856922, 749628208

Offset: 0

Alois P. Heinz, Nov 19 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Pentomino

Cf. A174249, A233427, A234931, A077909, A247127.

gf:= -(x+1) *(4*x^19 -4*x^18 +8*x^17 -4*x^16 +12*x^15 -12*x^14 +9*x^13 -5*x^12 -2*x^10 +5*x^9 -6*x^8 +10*x^7 -10*x^6 +8*x^5 -7*x^4 +4*x^3 -3*x^2 +3*x-1) / (4*x^23 +8*x^22 +12*x^21 +32*x^20 +8*x^19 +6*x^18 -15*x^17 -22*x^16 -9*x^15 -9*x^14 +13*x^13 +4*x^12 +22*x^11 -15*x^10 +x^9 -9*x^8 -x^7 +3*x^6 +3*x^5 +3*x^4 -2*x^3 -2*x+1):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..50);

G.f.: see Maple program.

A077976 Expansion of 1/(1+x+x^2+2*x^3).

Original entry on oeis.org

1, -1, 0, -1, 3, -2, 1, -5, 8, -5, 7, -18, 21, -17, 32, -57, 59, -66, 121, -173, 184, -253, 415, -530, 621, -921, 1360, -1681, 2163, -3202, 4401, -5525, 7528, -10805, 14327, -18578, 25861, -35937, 47232, -63017, 87659, -119106, 157481, -213693, 294424, -395693, 528655, -721810, 984541, -1320041

Offset: 0

N. J. A. Sloane, Nov 17 2002

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

Partial sums give: A077909.

a:=[1,-1,0];; for n in [4..50] do a[n]:=-a[n-1]-a[n-2]-2*a[n-3]; od; a; # G. C. Greubel, Jun 25 2019

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( 1/(1+x+x^2+2*x^3) )); // G. C. Greubel, Jun 25 2019

LinearRecurrence[{-1, -1, -2}, {1, -1, 0}, 50] (* Vladimir Joseph Stephan Orlovsky, Feb 24 2012 *)

my(x='x+O('x^50)); Vec(1/(1+x+x^2+2*x^3)) \\ G. C. Greubel, Jun 25 2019

(1/(1+x+x^2+2*x^3)).series(x, 50).coefficients(x, sparse=False) # G. C. Greubel, Jun 25 2019

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A128099 Triangle read by rows: T(n,k) is the number of ways to tile a 3 X n rectangle with k pieces of 2 X 2 tiles and 3n-4k pieces of 1 X 1 tiles (0 <= k <= floor(n/2)).

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A234312 Number of tilings of a 5 X n rectangle using n pentominoes of shapes L, X.

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A247126 Number of tilings of a 5 X n rectangle using n pentominoes of shapes F, U, X, N.

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A077976 Expansion of 1/(1+x+x^2+2*x^3).

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