cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A123521 Triangle read by rows: T(n,k)=number of tilings of a 2 X n grid with k pieces of 1 X 2 tiles (in horizontal position) and 2n-2k pieces of 1 X 1 tiles (0<=k<=n).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 4, 4, 1, 6, 11, 6, 1, 1, 8, 22, 24, 9, 1, 10, 37, 62, 46, 12, 1, 1, 12, 56, 128, 148, 80, 16, 1, 14, 79, 230, 367, 314, 130, 20, 1, 1, 16, 106, 376, 771, 920, 610, 200, 25, 1, 18, 137, 574, 1444, 2232, 2083, 1106, 295, 30, 1, 1, 20, 172, 832, 2486, 4744, 5776, 4352, 1897, 420, 36
Offset: 0

Views

Author

Emeric Deutsch, Oct 16 2006

Keywords

Comments

Also the triangle of the coefficients of the squares of the Fibonacci polynomials. Row n has 1+2*floor(n/2) terms. Sum of terms in row n = (Fibonacci(n+1))^2 (A007598).
From Michael A. Allen, Jun 24 2020: (Start)
T(n,k) is the number of tilings of an n-board (a board with dimensions n X 1) using k (1/2, 1/2)-fence tiles and 2*(n-k) half-squares (1/2 X 1 pieces, always placed so that the shorter sides are horizontal). A (1/2, 1/2)-fence is a tile composed of two 1/2 X 1 pieces separated by a gap of width 1/2.
T(n,k) is the (n, (n-k))-th entry of the (1/(1-x^2), x/(1-x)^2) Riordan array.
(-1)^(n+k)*T(n,k) is the (n, (n-k))-th entry of the (1/(1-x^2), x/(1+x)^2) Riordan array (A158454). (End)

Examples

			T(3,1)=4 because the 1 X 2 tile can be placed in any of the four corners of the 2 X 3 grid.
The irregular triangle begins as:
  1;
  1;
  1,  2,   1;
  1,  4,   4;
  1,  6,  11,   6,    1;
  1,  8,  22,  24,    9;
  1, 10,  37,  62,   46,   12,    1;
  1, 12,  56, 128,  148,   80,   16;
  1, 14,  79, 230,  367,  314,  130,   20,    1;
  1, 16, 106, 376,  771,  920,  610,  200,   25;
  1, 18, 137, 574, 1444, 2232, 2083, 1106,  295,  30,  1;
  1, 20, 172, 832, 2486, 4744, 5776, 4352, 1897, 420, 36;

References

  • Kenneth Edwards, Michael A. Allen, A new combinatorial interpretation of the Fibonacci numbers squared, Part II, Fib. Q., 58:2 (2020), 169-177.

Links

Crossrefs

Cf. A007598, A158454.
Other triangles related to tiling using fences: A059259, A157897, A335964.

Programs

  • Magma
    function A123521(n,k)
  if k eq 0 then return 1;
  elif k eq 1 then return 2*(n-1);
  else return A123521(n-2,k-2) + Binomial(2*n-k-1, 2*n-2*k-1);
  end if; return A123521;
end function;
[A123521(n,k): k in [0..2*Floor(n/2)], n in [0..14]]; // G. C. Greubel, Sep 01 2022
  • Maple
    G:=(1-t*z)/(1+t*z)/(1-z-2*t*z+t^2*z^2): Gser:=simplify(series(G,z=0,14)): for n from 0 to 11 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 11 do seq(coeff(P[n],t,k),k=0..2*floor(n/2)) od; # yields sequence in triangular form
  • Mathematica
    Block[{T}, T[0, 0]= T[1, 0]= 1; T[n_, k_]:= Which[k==0, 1, k==1, 2(n-1), True, T[n -2, k-2] + Binomial[2n-k-1, 2n-2k-1]]; Table[T[n, k], {n, 0, 14}, {k, 0, 2 Floor[n/2]}]] // Flatten (* Michael De Vlieger, Jun 24 2020 *)
  • SageMath
    @CachedFunction
def T(n,k): # T = A123521
    if (k==0): return 1
    elif (k==1): return 2*(n-1)
    else: return T(n-2, k-2) + binomial(2*n-k-1, 2*n-2*k-1)
flatten([[T(n,k) for k in (0..2*(n//2))] for n in (0..12)]) # G. C. Greubel, Sep 01 2022

Formula

G.f.: G = (1-t*z)/((1+t*z)*(1-z-2*t*z+t^2*z^2)). G = 1/(1-g), where g = z+t^2*z^2+2*t*z^2/(1-t*z) is the g.f. of the indecomposable tilings, i.e., of those that cannot be split vertically into smaller tilings. The row generating polynomials are P(n) = (Fibonacci(n))^2. They satisfy the recurrence relation P(n) = (1+t)*(P(n-1) + t*P(n-2)) - t^3*P(n-3).
T(n,k) = T(n-2,k-2) + binomial(2*n-k-1, 2*n-2*k-1). - Michael A. Allen, Jun 24 2020

A157897 Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k-1) + T(n-3,k-3) + delta(n,0)*delta(k,0), T(n,k<0) = T(n

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 1, 1, 3, 1, 2, 0, 1, 4, 3, 3, 2, 0, 1, 5, 6, 5, 6, 0, 1, 1, 6, 10, 9, 12, 3, 3, 0, 1, 7, 15, 16, 21, 12, 6, 3, 0, 1, 8, 21, 27, 35, 30, 14, 12, 0, 1, 1, 9, 28, 43, 57, 61, 35, 30, 6, 4, 0, 1, 10, 36, 65, 91, 111, 81, 65, 30, 10, 4, 0, 1, 11, 45, 94, 142, 189, 169, 135, 90, 30, 20, 0, 1
Offset: 0

Views

Author

Gary W. Adamson, Mar 08 2009

Keywords

Comments

T(n, k) is the number of tilings of an n-board that use k (1/2, 1)-fences and n-k squares. A (1/2, 1)-fence is a tile composed of two pieces of width 1/2 separated by a gap of width 1. (Result proved in paper by K. Edwards - see the links section.) - Michael A. Allen, Apr 28 2019
T(n, k) is the (n, n-k)-th entry in the (1/(1-x^3), x*(1+x)/(1-x^3)) Riordan array. - Michael A. Allen, Mar 11 2021

Examples

			First few rows of the triangle are:
  1;
  1,  0;
  1,  1,  0;
  1,  2,  0,  1;
  1,  3,  1,  2,  0;
  1,  4,  3,  3,  2,  0;
  1,  5,  6,  5,  6,  0,  1;
  1,  6, 10,  9, 12,  3,  3,  0;
  1,  7, 15, 16, 21, 12,  6,  3,  0;
  1,  8, 21, 27, 35, 30, 14, 12,  0,  1;
  ...
T(9,3) = 27 = T(8,3) + T(7,2) + T(6,0) = 16 + 10 + 1.

Links

Crossrefs

Cf. A000073 (row sums), A006498, A120415.
Cf. A001477, A079978, A087508, A120415, A161680.
Other triangles related to tiling using fences: A059259, A123521, A335964.

Programs

  • Magma
    function T(n,k) // T = A157897
  if k lt 0 or k gt n then return 0;
  elif k eq 0 then return 1;
  else return T(n-1, k) + T(n-2, k-1) + T(n-3, k-3);
  end if; return T;
end function;
[T(n,k): k in [0..n], n in [0..14]]; // G. C. Greubel, Sep 01 2022
  • Mathematica
    T[n_,k_]:= If[nMichael A. Allen, Apr 28 2019 *)
  • SageMath
    def T(n,k): # T = A157897
    if (k<0 or k>n): return 0
    elif (k==0): return 1
    else: return T(n-1, k) + T(n-2, k-1) + T(n-3, k-3)
flatten([[T(n,k) for k in (0..n)] for n in (0..14)]) # G. C. Greubel, Sep 01 2022

Formula

T(n,k) = T(n-1,k) + T(n-2,k-1) + T(n-3,k-3) + delta(n,0)*delta(k,0), T(n,k<0) = T(n
Sum_{k=0..n} T(n, k) = A000073(n+2). - Reinhard Zumkeller, Jun 25 2009
From G. C. Greubel, Sep 01 2022: (Start)
T(n, k) = T(n-1, k) + T(n-2, k-1) + T(n-3, k-3), with T(n, 0) = 1.
T(n, n) = A079978(n).
T(n, n-1) = A087508(n), n >= 1.
T(n, 1) = A001477(n-1).
T(n, 2) = A161680(n-2).
Sum_{k=0..floor(n/2)} T(n-k, k) = A120415(n). (End)

Extensions

Name clarified by Michael A. Allen, Apr 28 2019
Definition improved by Michael A. Allen, Mar 11 2021

A350110 Triangle read by rows, T(n,k) = T(n-1,k) + T(n-1,k-1) - T(n-2,k-1) + T(n-3,k-1) + T(n-3,k-2) + T(n-3,k-3) - T(n-4,k-3) - T(n-4,k-4) + delta(n,0)*delta(k,0) - delta(n,1)*delta(k,1), T(n

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 2, 3, 2, 0, 1, 3, 5, 4, 0, 0, 1, 4, 8, 8, 4, 2, 1, 1, 5, 12, 16, 13, 9, 3, 0, 1, 6, 17, 28, 30, 22, 9, 0, 0, 1, 7, 23, 45, 58, 51, 27, 9, 3, 1, 1, 8, 30, 68, 103, 108, 78, 40, 18, 4, 0, 1, 9, 38, 98, 171, 211, 187, 123, 58, 16, 0, 0
Offset: 0

Views

Author

Michael A. Allen, Dec 21 2021

Keywords

Comments

This is the m=3 member in the sequence of triangles A007318, A059259, A350110, A350111, A350112 which give the number of tilings of an (n+k) X 1 board using k (1,m-1)-fences and n-k unit square tiles. A (1,g)-fence is composed of two unit square tiles separated by a gap of width g.
It is also the m=3, t=2 member of a two-parameter family of triangles such that T(n,k) is the number of tilings of an (n+(t-1)*k) X 1 board using k (1,m-1;t)-combs and n-k unit square tiles. A (1,g;t)-comb is composed of a line of t unit square tiles separated from each other by gaps of width g. - Michael A. Allen, Dec 27 2021
T(3*j+r-k,k) is the coefficient of x^k in (f(j,x))^(3-r)*(f(j+1,x))^r for r=0,1,2 where f(n,x) is one form of a Fibonacci polynomial defined by f(n+1,x)=f(n,x)+x*f(n-1,x) where f(0,x)=1 and f(n<0,x)=0.
T(n+3-k,k) is the number of subsets of {1,2,...,n} of size k such that no two elements in a subset differ by 3.
Sum of (n+3)-th antidiagonal (counting initial 1 as the 0th) is A006500(n).

Examples

			Triangle begins:
   1;
   1,   0;
   1,   0,   0;
   1,   1,   1,   1;
   1,   2,   3,   2,   0;
   1,   3,   5,   4,   0,   0;
   1,   4,   8,   8,   4,   2,   1;
   1,   5,  12,  16,  13,   9,   3,   0;
   1,   6,  17,  28,  30,  22,   9,   0,   0;
   1,   7,  23,  45,  58,  51,  27,   9,   3,   1;
   1,   8,  30,  68, 103, 108,  78,  40,  18,   4,   0;
   1,   9,  38,  98, 171, 211, 187, 123,  58,  16,   0,   0;
   1,  10,  47, 136, 269, 382, 399, 310, 176,  64,  16,   4,   1;

Links

Crossrefs

Other members of the two-parameter family of triangles: A007318 (m=1,t=2), A059259 (m=2,t=2), A350111 (m=4,t=2), A350112 (m=5,t=2), A354665 (m=2,t=3), A354666 (m=2,t=4), A354667 (m=2,t=5), A354668 (m=3,t=3).
Other triangles related to tiling using fences: A123521, A157897, A335964.

Programs

  • Mathematica
    T[n_, k_]:=If[k<0 || n

Formula

T(n,0) = 1.
T(n,n) = delta(n mod 3,0).
T(n,1) = n-2 for n>1.
T(3*j-r,3*j-p) = 0 for j>0, p=1,2, and r=1,...,p.
T(3*(j-1)+p,3*(j-1)) = T(3*j,3*j-p) = j^p for j>0 and p=0,1,2,3.
T(3*j+1,3*j-1) = 3*j(j+1)/2 for j>0.
T(3*j+2,3*j-2) = 3*(C(j+2,4) + C(j+1,2)^2) for j>1.
G.f. of row sums: (1-x)/((1-2*x)*(1+x^2-x^3)).
G.f. of antidiagonal sums: (1-x^2)/((1-x-x^2)*(1+x^3-x^6)).
T(n,k) = T(n-1,k) + T(n-1,k-1) for n>=2*k+1 if k>=0.

A350111 Triangle read by rows: T(n,k) is the number of tilings of an (n+k)-board using k (1,3)-fences and n-k squares.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 3, 4, 2, 0, 1, 3, 6, 7, 4, 0, 0, 1, 4, 9, 12, 8, 0, 0, 0, 1, 5, 13, 20, 16, 8, 4, 2, 1, 1, 6, 18, 32, 36, 28, 19, 12, 3, 0, 1, 7, 24, 50, 69, 69, 58, 31, 9, 0, 0, 1, 8, 31, 74, 120, 144, 127, 78, 27, 0, 0, 0
Offset: 0

Views

Author

Michael A. Allen, Dec 22 2021

Keywords

Comments

This is the m=4 member in the sequence of triangles A007318, A059259, A350110, A350111, A350112 which give the number of tilings of an (n+k) X 1 board using k (1,m-1)-fences and n-k unit square tiles. A (1,g)-fence is composed of two unit square tiles separated by a gap of width g.
It is also the m=4, t=2 member of a two-parameter family of triangles such that T(n,k) is the number of tilings of an (n+(t-1)*k) X 1 board using k (1,m-1;t)-combs and n-k unit square tiles. A (1,g;t)-comb is composed of a line of t unit square tiles separated from each other by gaps of width g.
T(4*j+r-k,k) is the coefficient of x^k in (f(j,x))^(4-r)*(f(j+1,x))^r for r=0,1,2,3 where f(n,x) is one form of a Fibonacci polynomial defined by f(n+1,x)=f(n,x)+x*f(n-1,x) where f(0,x)=1 and f(n<0,x)=0.
T(n+4-k,k) is the number of subsets of {1,2,...,n} of size k such that no two elements in a subset differ by 4.
Sum of (n+3)-th antidiagonal (counting initial 1 as the 0th) is A031923(n).

Examples

			Triangle begins:
  1;
  1,   0;
  1,   0,   0;
  1,   0,   0,   0;
  1,   1,   1,   1,   1;
  1,   2,   3,   4,   2,   0;
  1,   3,   6,   7,   4,   0,   0;
  1,   4,   9,  12,   8,   0,   0,   0;
  1,   5,  13,  20,  16,   8,   4,   2,   1;
  1,   6,  18,  32,  36,  28,  19,  12,   3,   0;
  1,   7,  24,  50,  69,  69,  58,  31,   9,   0,   0;
  1,   8,  31,  74, 120, 144, 127,  78,  27,   0,   0,   0;
  1,   9,  39, 105, 195, 264, 265, 189,  81,  27,   9,   3,   1;
  1,  10,  48, 144, 300, 458, 522, 432, 270, 132,  58,  24,   4,   0;

Links

Crossrefs

Sums of antidiagonals: A031923
Other members of the two-parameter family of triangles: A007318 (m=1,t=2), A059259 (m=2,t=2), A350110 (m=3,t=2), A350112 (m=5,t=2), A354665 (m=2,t=3), A354666 (m=2,t=4), A354667 (m=2,t=5), A354668 (m=3,t=3).
Other triangles related to tiling using fences: A123521, A157897, A335964.

Programs

  • Mathematica
    f[n_]:=If[n<0,0,f[n-1]+x*f[n-2]+KroneckerDelta[n,0]];
T[n_, k_]:=Module[{j=Floor[(n+k)/4],r=Mod[n+k,4]},
  Coefficient[f[j]^(4-r)*f[j+1]^r,x,k]];
Flatten@Table[T[n,k], {n, 0, 13}, {k, 0, n}]
(* or *)
T[n_,k_]:=If[k<0 || n

Formula

T(n,k) = T(n-1,k) + T(n-2,k-1) - T(n-3,k-1) + T(n-3,k-2) + T(n-4,k-1) + T(n-4,k-3) + 2*T(n-4,k-4) + T(n-5,k-2) + 2*T(n-5,k-3) - T(n-5,k-4) - T(n-6,k-3)-T(n-6,k-5) - T(n-7,k-4)-T(n-7,k-5) - T(n-7,k-6) - T(n-8,k-7)-T(n-8,k-8) + delta(n,0)*delta(k,0) - delta(n,2)*delta(k,1) - delta(n,3)*delta(k,2) - delta(n,4)*delta(k,4) with T(n
T(n,0) = 1.
T(n,n) = delta(n mod 4,0).
T(n,1) = n-3 for n>2.
T(4*j-r,4*j-p) = 0 for j>0, p=1,2,3, and r=1,...,p.
T(4*(j-1)+p,4*(j-1)) = T(4*j,4*j-p) = j^p for j>0 and p=0,1,2,3,4.
T(4*j+1,4*j-1) = 4*j(j+1)/2 for j>0.
T(4*j+2,4*j-2) = 4*C(j+2,4) + 6*C(j+1,2)^2 for j>1.
G.f. of row sums: (1-x-x^3)/((1-2*x)*(1-x^2)*(1+2*x^2+x^3+x^4)).
G.f. of antidiagonal sums: (1-x^2-x^3+x^4-x^6)/((1-x-x^2)*(1-x^4)*(1+3*x^4+x^8)).
T(n,k) = T(n-1,k) + T(n-1,k-1) for n>=3*k+1 if k>=0.

A350112 Triangle read by rows: T(n,k) is the number of tilings of an (n+k)-board using k (1,4)-fences and n-k squares.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 2, 0, 1, 3, 6, 10, 9, 4, 0, 0, 1, 4, 10, 16, 16, 8, 0, 0, 0, 1, 5, 14, 25, 28, 16, 0, 0, 0, 0, 1, 6, 19, 38, 48, 32, 16, 8, 4, 2, 1, 1, 7, 25, 56, 80, 80, 60, 40, 25, 15, 3, 0, 1, 8, 32, 80, 136, 166, 157, 128, 95, 40, 9, 0, 0
Offset: 0

Views

Author

Michael A. Allen, Dec 22 2021

Keywords

Comments

This is the m=5 member in the sequence of triangles A007318, A059259, A350110, A350111, A350112 which give the number of tilings of an (n+k) X 1 board using k (1,m-1)-fences and n-k unit square tiles. A (1,g)-fence is composed of two unit square tiles separated by a gap of width g.
It is also the m=5, t=2 member of a two-parameter family of triangles such that T(n,k) is the number of tilings of an (n+(t-1)*k) X 1 board using k (1,m-1;t)-combs and n-k unit square tiles. A (1,g;t)-comb is composed of a line of t unit square tiles separated from each other by gaps of width g.
T(5*j+r-k,k) is the coefficient of x^k in (f(j,x))^(5-r)*(f(j+1,x))^r for r=0,1,2,3,4 where f(n,x) is one form of a Fibonacci polynomial defined by f(n+1,x)=f(n,x)+x*f(n-1,x) where f(0,x)=1 and f(n<0,x)=0.
T(n+5-k,k) is the number of subsets of {1,2,...,n} of size k such that no two elements in a subset differ by 5.
Sum of (5j+r)-th antidiagonal (counting initial 1 as the 0th) is f(j)^(5-r)*f(j+1)^r where j=0,1,..., r=0,1,2,3,4, and f(n) is the Fibonacci number A000045(n+1).

Examples

			Triangle begins:
  1;
  1,   0;
  1,   0,   0;
  1,   0,   0,   0;
  1,   0,   0,   0,   0;
  1,   1,   1,   1,   1,   1;
  1,   2,   3,   4,   5,   2,   0;
  1,   3,   6,  10,   9,   4,   0,   0;
  1,   4,  10,  16,  16,   8,   0,   0,   0;
  1,   5,  14,  25,  28,  16,   0,   0,   0,   0;
  1,   6,  19,  38,  48,  32,  16,   8,   4,   2,   1;
  1,   7,  25,  56,  80,  80,  60,  40,  25,  15,   3,   0;
  1,   8,  32,  80, 136, 166, 157, 128,  95,  40,   9,   0,   0;
  1,   9,  40, 112, 217, 309, 346, 330, 223, 105,  27,   0,   0,   0;

Links

Crossrefs

Other members of the two-parameter family of triangles: A007318 (m=1,t=2), A059259 (m=2,t=2), A350110 (m=3,t=2), A350111 (m=4,t=2), A354665 (m=2,t=3), A354666 (m=2,t=4), A354667 (m=2,t=5), A354668 (m=3,t=3).
Other triangles related to tiling using fences: A123521, A157897, A335964.

Programs

  • Mathematica
    f[n_]:=If[n<0,0,f[n-1]+x*f[n-2]+KroneckerDelta[n,0]];
T[n_, k_]:=Module[{j=Floor[(n+k)/5], r=Mod[n+k,5]},
  Coefficient[f[j]^(5-r)*f[j+1]^r,x,k]];
Flatten@Table[T[n,k], {n, 0, 13}, {k, 0, n}]

Formula

T(n,0) = 1.
T(n,n) = delta(n mod 5,0).
T(n,1) = n-4 for n>3.
T(5*j-r,5*j-p) = 0 for j>0, p=1,2,3,4, and r=1,...,p.
T(5*(j-1)+p,5*(j-1)) = T(5*j,5*j-p) = j^p for j>0 and p=0,1,...,5.
T(5*j+1,5*j-1) = 5*j(j+1)/2 for j>0.
T(5*j+2,5*j-2) = 5*C(j+2,4) + 10*C(j+1,2)^2 for j>1.
T(n,k) = T(n-1,k) + T(n-1,k-1) for n >= 4*k+1 if k >= 0.

A158909 Riordan array (1/((1-x)(1-x^2)), x/(1-x)^2). Triangle read by rows, T(n,k) for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 2, 7, 5, 1, 3, 13, 16, 7, 1, 3, 22, 40, 29, 9, 1, 4, 34, 86, 91, 46, 11, 1, 4, 50, 166, 239, 174, 67, 13, 1, 5, 70, 296, 553, 541, 297, 92, 15, 1, 5, 95, 496, 1163, 1461, 1068, 468, 121, 17, 1, 6, 125, 791, 2269, 3544, 3300, 1912, 695, 154, 19, 1
Offset: 0

Views

Author

Paul Barry, Mar 30 2009

Keywords

Comments

Diagonal sums are the Jacobsthal numbers A001045.
Transforms r^n into the symmetric third-order sequence with g.f. 1/(1-(r+1)x-(r+1)x^2+x^3), see the formulas.
From Wolfdieter Lang, Oct 22 2019: (Start)
The signed triangle t(n, k) = (-1)^(n-k)*T(n, k) appears in the expansion [n+2, 2]q / q^n = Sum{k=0} t(n, k)*y^(2*k), with y = q^(1/2) + q^(-1/2), where [n+2, 2]_q are q-binomial coefficients (see A008967, but with a different offset). The formula is [n+2, 2]_q / q^n = S(n+1, y)*S(n, y)/y with Chebyshev S polynomials (A049310). This is a polynomial in y^2 but not in q after replacement of the given y = y(q).
The A-sequence for this Riordan triangle is A(n) = (-1)^n*A115141(n) with o.g.f A(x) = 1 + x*(1 + c(-x)), with c(x) generating A000108 (Catalan).
The Z-sequence is z(n) = (-1)^(n+1)*A071724(n), for n >= 1 and z(0) = 1. The o.g.f. is Z(x) = 1 + x*c(-x)^3. See A071724 for a link on A- and Z-sequences, and their use for the recurrence. (End)
T(n,k) is the number of tilings of a (2*n+1)-board (a 1 X (2*n+1) rectangular board) using 2*k+1 squares and 2*(n-k) (1,1)-fences. A (1,1)-fence is a tile composed of two squares separated by a gap of width 1. - Michael A. Allen, Mar 20 2021

Examples

			From _Wolfdieter Lang_, Oct 22 2019: (Start)
The triangle T(n, k) begins:
  n\k  0   1   2    3    4    5    6   7   8  9 10 ...
  ----------------------------------------------------
  0:   1
  1:   1   1
  2:   2   3   1
  3:   2   7   5    1
  4:   3  13  16    7    1
  5:   3  22  40   29    9    1
  6:   4  34  86   91   46   11    1
  7:   4  50 166  239  174   67   13   1
  8:   5  70 296  553  541  297   92  15   1
  9:   5  95 496 1163 1461 1068  468 121  17  1
  10:  6 125 791 2269 3544 3300 1912 695 154 19  1
  ...
----------------------------------------------------------------------------
Recurrence: T(5, 2) = 16 + 13 + 5 + 7 - 1 = 40, and T(5, 0) = 3 + 2 - 2 = 3. [using _Philippe Deléham_'s Nov 12 2013 recurrence]
Recurrence from A-sequence [1, 2, -1, 2, -5, ...]: T(5, 2) = 1*13 + 2*16 - 1*7 + 2*1 = 40.
Recurrence from Z-sequence [1, 1, -3, 9, -28, ...]: T(5, 0) = 1*3 + 1*13 - 3*16 + 9*7 - 28*1 = 3. (End)

Links

Crossrefs

Cf. A000045, A001654 (row sums), A008967, A035317, A049310, A092879, A335964.

Programs

  • Magma
    [(&+[(-1)^(j+n-k)*Binomial(2*k+j+1, j): j in [0..n-k]]): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 18 2021
  • Maple
    T := (n,k) -> binomial(k+n+2, n-k+1)*hypergeom([1, k+n+3], [n-k+2], -1) + (-1)^(n-k)/4^(k+1):
seq(seq(simplify(T(n,k)), k=0..n), n=0..9); # Peter Luschny, Oct 31 2019
  • Mathematica
    Table[Sum[(-1)^(j+n-k)*Binomial[j+2*k+1, j], {j,0,n-k}], {n,0,12}, {k,0,n}] // Flatten (* G. C. Greubel, Mar 18 2021 *)
  • Sage
    flatten([[sum((-1)^(j+n-k)*binomial(j+2*k+1, j) for j in (0..n-k)) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 18 2021

Formula

Sum_{k=0..n} T(n,k) = Fibonacci(n+1)*Fibonacci(n+2) = A001654(n+1).
From Johannes W. Meijer, Jul 20 2011: (Start)
T(n, k) = Sum_{i=0..n-k} (-1)^(i+n-k) * binomial(i+2*k+1, i).
T(n, k) = A035317(n+k, n-k) = A092879(n, n-k).
Sum_{k=0..n} T(n, k)*r^k = coeftayl(1/(1-(r+1)*x-(r+1)*x^2+x^3), x=0, n). [Barry] (End)
T(n, k) = T(n-1, k) + T(n-1, k-1) + T(n-2, k) + T(n-2, k-1) - T(n-3, k), T(0, 0) = 1, T(n, k) = 0 if k<0 or if k>n. - Philippe Deléham, Nov 12 2013
From Wolfdieter Lang, Oct 22 2019: (Start)
O.g.f. for the row polynomials (that is for the triangle): G(z, x) = 1/((1 + z)*(1 - (x + 2)*z + z^2)), and
O.g.f. for column k: x^k/((1+x)*(1-x)^(2*(k+1))) (Riordan property). (End)
T(n, k) = binomial(k + n + 2, n - k + 1)*hypergeom([1, k + n + 3], [n - k + 2], -1) + (-1)^(n - k)/4^(k + 1). - Peter Luschny, Oct 31 2019
From Michael A. Allen, Mar 20 2021: (Start)
T(n,k) = A335964(2*n+1,n-k).
T(n,k) = T(n-2,k) + binomial(n+k,2*k). (End)

A354665 Triangle read by rows, T(n,k) = T(n-1,k) + T(n-1,k-1) - T(n-2,k-1) + T(n-2,k-2) + T(n-3,k-1) - T(n-3,k-3) + delta(n,0)*delta(k,0) - delta(n,1)*delta(k,1), T(n

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 1, 1, 2, 0, 1, 2, 4, 0, 1, 1, 3, 6, 3, 3, 0, 1, 4, 9, 8, 9, 0, 1, 1, 5, 13, 17, 18, 6, 4, 0, 1, 6, 18, 30, 36, 20, 16, 0, 1, 1, 7, 24, 48, 66, 55, 40, 10, 5, 0, 1, 8, 31, 72, 114, 120, 100, 40, 25, 0, 1, 1, 9, 39, 103, 186
Offset: 0

Views

Author

Michael A. Allen, Jun 04 2022

Keywords

Comments

This is the m=2, t=3 member of a two-parameter family of triangles such that T(n,k) is the number of tilings of an (n+(t-1)*k) X 1 board using k (1,m-1;t)-combs and n-k unit square tiles. A (1,g;t)-comb is composed of a line of t unit square tiles separated from each other by gaps of width g.
T(2*j+r-2*k,k) is the coefficient of x^k in (f(j,x))^(2-r)*(f(j+1,x))^r for r=0,1, where f(n,x) is a Narayana's cows polynomial defined by f(n,x)=f(n-1,x)+x*f(n-3,x)+delta(n,0) where f(n<0,x)=0.
T(n+4-2*k,k) is the number of subsets of {1,2,...,n} of size k such that no two elements in a subset differ by 2 or 4.

Examples

			Triangle begins:
  1;
  1,   0;
  1,   0,   1;
  1,   1,   2,   0;
  1,   2,   4,   0,   1;
  1,   3,   6,   3,   3,   0;
  1,   4,   9,   8,   9,   0,   1;
  1,   5,  13,  17,  18,   6,   4,   0;
  1,   6,  18,  30,  36,  20,  16,   0,   1;
  1,   7,  24,  48,  66,  55,  40,  10,   5,   0;
  1,   8,  31,  72, 114, 120, 100,  40,  25,   0,   1;
  1,   9,  39, 103, 186, 234, 221, 135,  75,  15,   6,   0;
...

Links

Crossrefs

Row sums are A011782.
Sums over k of T(n-2*k,k) are A224809.
Other members of the family of triangles: A007318 (m=1,t=2), A059259 (m=2,t=2), A350110 (m=3,t=2), A350111 (m=4,t=2), A350112 (m=5,t=2), A354666 (m=2,t=4), A354667 (m=2,t=5), A354668 (m=3,t=3).
Other triangles related to tiling using combs: A059259, A123521, A157897, A335964.

Programs

  • Mathematica
    T[n_, k_]:=If[k<0 || n

Formula

T(n,0) = 1.
T(n,n) = delta(n mod 2,0).
T(n,1) = n-2 for n>1.
T(2*j-r,2*j-1) = 0 for j>0, r=0,1.
T(2*(j-1)+p,2*(j-1)) = j^p for j>0 and p=0,1,2.
T(2*(j-1)+3,2*(j-1)) = j^2*(j+1)/2 for j>0.
T(2*j+p,2*j-p) = C(j+1,2)^p for j>0 and p=0,1,2.
G.f. of row sums: (1-x)/(1-2*x).
G.f. of sums of T(n-2*k,k) over k: (1-x^3)/((1-x-x^3)*(1+x^4-x^6)).
T(n,k) = T(n-1,k) + T(n-1,k-1) for n>=2*k+1 if k>=0.

A354666 Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k-1) + 2*T(n-2,k-2) - T(n-3,k-1) - T(n-3,k-2) + T(n-4,k-1) + T(n-4,k-2) - T(n-4,k-3) - T(n-4,k-4) + delta(n,0)*delta(k,0) - delta(n,2)*(delta(k,1) + delta(k,2)), T(n

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 1, 0, 2, 0, 1, 1, 4, 0, 1, 1, 2, 6, 0, 3, 0, 1, 3, 9, 4, 9, 0, 1, 1, 4, 12, 10, 18, 0, 4, 0, 1, 5, 16, 21, 36, 10, 16, 0, 1, 1, 6, 21, 36, 60, 30, 40, 0, 5, 0, 1, 7, 27, 57, 100, 81, 100, 20, 25, 0, 1, 1, 8, 34, 84, 158, 168
Offset: 0

Views

Author

Michael A. Allen, Jun 04 2022

Keywords

Comments

This is the m=2, t=4 member of a two-parameter family of triangles such that T(n,k) is the number of tilings of an (n+(t-1)*k) X 1 board using k (1,m-1;t)-combs and n-k unit square tiles. A (1,g;t)-comb is composed of a line of t unit square tiles separated from each other by gaps of width g.
T(2*j+r-3*k,k) is the coefficient of x^k in (f(j,x))^(2-r)*(f(j+1,x))^r for r=0,1, where f(n,x) is a (1,4)-bonacci polynomial defined by f(n,x)=f(n-1,x)+x*f(n-4,x)+delta(n,0) where f(n<0,x)=0.
T(n+6-3*k,k) is the number of subsets of {1,2,...,n} of size k such that no two elements in a subset differ by 2, 4, or 6.

Examples

			Triangle begins:
  1;
  1,   0;
  1,   0,   1;
  1,   0,   2,   0;
  1,   1,   4,   0,   1;
  1,   2,   6,   0,   3,   0;
  1,   3,   9,   4,   9,   0,   1;
  1,   4,  12,  10,  18,   0,   4,   0;
  1,   5,  16,  21,  36,  10,  16,   0,   1;
  1,   6,  21,  36,  60,  30,  40,   0,   5,   0;
  1,   7,  27,  57, 100,  81, 100,  20,  25,   0,   1;
  1,   8,  34,  84, 158, 168, 200,  70,  75,   0,   6,   0;
  1,   9,  42, 118, 243, 322, 400, 231, 225,  35,  36,   0,   1;
...

Links

Crossrefs

Row sums are A099163.
Sums over k of T(n-3*k,k) are A224808.
Other members of the family of triangles: A007318 (m=1,t=2), A059259 (m=2,t=2), A350110 (m=3,t=2), A350111 (m=4,t=2), A350112 (m=5,t=2), A354665 (m=2,t=3), A354667 (m=2,t=5), A354668 (m=3,t=3).
Other triangles related to tiling using combs: A059259, A123521, A157897, A335964.

Programs

  • Mathematica
    T[n_,k_]:=If[k<0 || n

Formula

T(n,0) = 1.
T(n,n) = delta(n mod 2,0).
T(n,1) = n-3 for n>2.
T(2*j-r,2*j-1) = 0 for j>0, r=-1,0,1.
T(2*(j-1)+p,2*(j-1)) = j^p for j>0 and p=0,1,2.
T(2*j+p,2*(j-1)) = j^2*((j+1)/2)^p for j>0 and p=1,2.
T(2*j+3,2*(j-1)) = (j*(j+1))^2*(j+2)/12 for j>0.
T(2*(j+p),2*j-p) = C(j+2,3)^p for j>0 and p=0,1,2.
G.f. of row sums: (1-2*x^2)/(1-x-3*x^2+2*x^3).
G.f. of sums of T(n-3*k,k) over k: (1-x^5-x^8)/(1-x-x^5+x^6-x^7-2*x^8+x^9-x^10+x^13+x^16).
T(n,k) = T(n-1,k) + T(n-1,k-1) for n>=3*k+1 if k>=0.

A354667 Triangle read by rows: T(n,k) is the number of tilings of an (n+4*k) X 1 board using k (1,1;5)-combs and n-k squares.

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 1, 0, 2, 0, 1, 0, 4, 0, 1, 1, 1, 6, 0, 3, 0, 1, 2, 9, 0, 9, 0, 1, 1, 3, 12, 5, 18, 0, 4, 0, 1, 4, 16, 12, 36, 0, 16, 0, 1, 1, 5, 20, 25, 60, 15, 40, 0, 5, 0, 1, 6, 25, 42, 100, 42, 100, 0, 25, 0, 1, 1, 7, 31, 66, 150, 112, 200
Offset: 0

Views

Author

Michael A. Allen, Jun 05 2022

Keywords

Comments

This is the m=2, t=5 member of a two-parameter family of triangles such that T(n,k) is the number of tilings of an (n+(t-1)*k) X 1 board using k (1,m-1;t)-combs and n-k unit square tiles. A (1,g;t)-comb is composed of a line of t unit square tiles separated from each other by gaps of width g.
T(2*j+r-4*k,k) is the coefficient of x^k in (f(j,x))^(2-r)*(f(j+1,x))^r for r=0,1, where f(n,x) is a (1,5)-bonacci polynomial defined by f(n,x)=f(n-1,x)+x*f(n-5,x)+delta(n,0) where f(n<0,x)=0.
T(n+8-4*k,k) is the number of subsets of {1,2,...,n} of size k such that no two elements in a subset differ by 2, 4, 6, or 8.

Examples

			Triangle begins:
  1;
  1,   0;
  1,   0,   1;
  1,   0,   2,   0;
  1,   0,   4,   0,   1;
  1,   1,   6,   0,   3,   0;
  1,   2,   9,   0,   9,   0,   1;
  1,   3,  12,   5,  18,   0,   4,   0;
  1,   4,  16,  12,  36,   0,  16,   0,   1;
  1,   5,  20,  25,  60,  15,  40,   0,   5,   0;
  1,   6,  25,  42, 100,  42, 100,   0,  25,   0,   1;
  1,   7,  31,  66, 150, 112, 200,  35,  75,   0,   6,   0;
...

Links

Crossrefs

Row sums are A005578.
Sums over k of T(n-4*k,k) are A224811.
Other members of the family of triangles: A007318 (m=1,t=2), A059259 (m=2,t=2), A350110 (m=3,t=2), A350111 (m=4,t=2), A350112 (m=5,t=2), A354665 (m=2,t=3), A354666 (m=2,t=4), A354668 (m=3,t=3).
Other triangles related to tiling using combs: A059259, A123521, A157897, A335964.

Programs

  • Mathematica
    T[n_,k_]:=If[k<0 || n

Formula

T(n,k) = T(n-1,k) + T(n-1,k-1) - T(n-2,k-1) + 2*T(n-2,k-2) + T(n-3,k-1) - T(n-3,k-2) - 2*T(n-3,k-3) - T(n-4,k-1) + T(n-4,k-2) + T(n-4,k-3) - T(n-4,k-4) + T(n-5,k-1) - 2*T(n-5,k-3) + T(n-5,k-5) + delta(n,0)*delta(k,0) - delta(n,1)*delta(k,1) - delta(n,2)*delta(k,2) - delta(n,3)*(delta(k,1) - delta(k,3)) with T(n,k<0) = T(n
T(n,0) = 1.
T(n,n) = delta(n mod 2,0).
T(n,1) = n-4 for n>3.
T(2*j+r,2*j-1) = 0 for j>0, r=-1,0,1,2.
T(n,2*j) = C(n/2,j)^2 for j>0 and n even and 2*j <= n <= 2*j+8.
T(n,2*j) = C((n-1)/2,j)*C((n+1)/2,j) for j>0 and n odd and 2*j < n < 2*j+8.
T(2*j+3*p,2*j-p) = C(j+3,4)^p for j>0 and p=0,1,2.
G.f. of row sums: (1-x-x^2)/(1-2*x-x^2+2*x^3).
G.f. of sums of T(n-4*k,k) over k: (1-x^5-x^7-x^10+x^15)/(1-x-x^5+x^6-x^7+x^8-x^9-2*x^10+x^11-x^12+2*x^15-x^16+2*x^17+x^20-x^25).
T(n,k) = T(n-1,k) + T(n-1,k-1) for n>=4*k+1 if k>=0.

A354668 Triangle read by rows: T(n,k) is the number of tilings of an (n+2*k) X 1 board using k (1,2;3)-combs and n-k squares.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 2, 0, 1, 1, 3, 4, 0, 0, 1, 2, 5, 8, 0, 0, 1, 1, 3, 8, 12, 0, 3, 3, 0, 1, 4, 12, 18, 9, 12, 9, 0, 0, 1, 5, 16, 27, 25, 29, 27, 0, 0, 1, 1, 6, 21, 42, 51, 66, 54, 0, 6, 4, 0, 1, 7, 27, 62, 95, 135, 108, 36
Offset: 0

Views

Author

Michael A. Allen, Jul 30 2022

Keywords

Comments

This is the m=3, t=3 member of a two-parameter family of triangles such that T(n,k) is the number of tilings of an (n+(t-1)*k) X 1 board using k (1,m-1;t)-combs and n-k unit square tiles. A (1,g;t)-comb is composed of a line of t unit square tiles separated from each other by gaps of width g.
T(3*j+r-2*k,k) is the coefficient of x^k in (f(j,x))^(3-r)*(f(j+1,x))^r for r=0,1, where f(n,x) is a Narayana's cows polynomial defined by f(n,x)=f(n-1,x)+x*f(n-3,x)+delta(n,0) where f(n<0,x)=0.
T(n+6-2*k,k) is the number of subsets of {1,2,...,n} of size k such that no two elements in a subset differ by 3 or 6.

Examples

			Triangle begins:
  1;
  1,   0;
  1,   0,   0;
  1,   0,   0,   1;
  1,   0,   1,   2,   0;
  1,   1,   3,   4,   0,   0;
  1,   2,   5,   8,   0,   0,   1;
  1,   3,   8,  12,   0,   3,   3,   0;
  1,   4,  12,  18,   9,  12,   9,   0,   0;
  1,   5,  16,  27,  25,  29,  27,   0,   0,   1;
  1,   6,  21,  42,  51,  66,  54,   0,   6,   4,   0;
  1,   7,  27,  62,  95, 135, 108,  36,  30,  16,   0,   0;
...

Links

Crossrefs

Sums over k of T(n-2*k,k) are A224810.
Other members of the family of triangles: A007318 (m=1,t=2), A059259 (m=2,t=2), A350110 (m=3,t=2), A350111 (m=4,t=2), A350112 (m=5,t=2), A354665 (m=2,t=3), A354666 (m=2,t=4), A354667 (m=2,t=5).
Other triangles related to tiling using combs: A059259, A123521, A157897, A335964.

Programs

  • Mathematica
    f[n_]:=If[n<0, 0, f[n-1]+x*f[n-3]+KroneckerDelta[n,0]]; T[n_, k_]:=Module[{j=Floor[(n+2*k)/3], r=Mod[n+2*k,3]}, Coefficient[f[j]^(3-r)*f[j+1]^r, x, k]]; Flatten@Table[T[n,k], {n, 0, 11}, {k, 0, n}]

Formula

T(n,0) = 1.
T(n,n) = delta(n mod 3,0).
T(n,1) = n-4 for n>3.
T(3*j-r,3*j-p) = 0 for j>0, p=1,2, and r=1-p,...,p.
T(n,2*j) = C(n/2,j)^2 for j>0 and n even and 2*j <= n <= 2*j+8.
T(n,2*j) = C((n-1)/2,j)*C((n+1)/2,j) for j>0 and n odd and 2*j < n < 2*j+8.
T(2*j+3*p,2*j-p) = C(j+3,4)^p for j>0 and p=0,1,2.
G.f. of sums of T(n-2*k,k) over k: (1+x^3-x^4-x^5+x^6-2*x^7-x^8-x^9-2*x^10-x^12-x^13-x^15)/((1-x)*(1+x+x^2)*(1-x-x^3)*(1+3*x^3+7*x^6+9*x^9+7*x^12+3*x^15+x^18)).
T(n,k) = T(n-1,k) + T(n-1,k-1) for n>=4*k+1 if k>=0.
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