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

Author

Emeric Deutsch, Feb 18 2007

Keywords

• nonn
• tabf

Comments

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

Examples

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

References

• 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

Links

• 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

Crossrefs

Cf. (Triangle sums) A001045, A095977, A077949, A052947, A113726, A052942, A077909.

Cf. (Similar triangles) A008315, A011973, A102541.

Cf. A013609, A038207

Cf. A207538

Programs

Maple

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

Mathematica

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 *)

Formula

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)