A348445 -id:A348445 - cp's OEIS frontend

A335964 Triangle read by rows, T(n,k) = T(n-1,k) + T(n-3,k-1) + T(n-4,k-2) + delta(n,0)*delta(k,0), T(n,k<0) = T(n

1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 1, 3, 2, 0, 0, 0, 1, 4, 4, 0, 0, 0, 0, 1, 5, 7, 2, 0, 0, 0, 0, 1, 6, 11, 6, 1, 0, 0, 0, 0, 1, 7, 16, 13, 3, 0, 0, 0, 0, 0, 1, 8, 22, 24, 9, 0, 0, 0, 0, 0, 0, 1, 9, 29, 40, 22, 3, 0, 0, 0, 0, 0, 0

Offset: 0

Michael A. Allen, Jul 01 2020

T(n,k) is the number of tilings of an n-board (a board with dimensions n X 1) using k (1,1)-fence tiles and n-2k square tiles. A (w,g)-fence tile is composed of two tiles of width w separated by a gap of width g.
Sum of n-th row = A006498(n).
T(2*j+r,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 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. - Michael A. Allen, Oct 02 2021

			Triangle begins:
  1;
  1,  0;
  1,  0,  0;
  1,  1,  0,  0;
  1,  2,  1,  0,  0;
  1,  3,  2,  0,  0,  0;
  1,  4,  4,  0,  0,  0,  0;
  1,  5,  7,  2,  0,  0,  0,  0;
  1,  6, 11,  6,  1,  0,  0,  0,  0;
  1,  7, 16, 13,  3,  0,  0,  0,  0,  0;
  1,  8, 22, 24,  9,  0,  0,  0,  0,  0,  0;
  1,  9, 29, 40, 22,  3,  0,  0,  0,  0,  0,  0;
  ...

Kenneth Edwards and Michael A. Allen, New Combinatorial Interpretations of the Fibonacci Numbers Squared, Golden Rectangle Numbers, and Jacobsthal Numbers Using Two Types of Tile, arXiv:2009.04649 [math.CO], 2020.
Kenneth Edwards and Michael A. Allen, New combinatorial interpretations of the Fibonacci numbers squared, golden rectangle numbers, and Jacobsthal numbers using two types of tile, J. Int. Seq. 24 (2021) Article 21.3.8.
John Konvalina, On the number of combinations without unit separation., Journal of Combinatorial Theory, Series A 31.2 (1981): 101-107. See Table I.

Other triangles related to tiling using fences: A059259, A123521, A157897, A158909.

Cf. A006498 (row sums), A011973, A348445.

Mathematica
```
T[n_,k_]:=If[n
				
```

TT(n,k) = if (nA059259
T(n,k) = TT(n-k,k);
\\ matrix(7,7,n,k, T(n-1,k-1)) \\ Michel Marcus, Jul 18 2020

T(n,k) = A059259(n-k,k).
From Michael A. Allen, Oct 02 2021: (Start)
G.f.: 1/((1 + x^2*y)(1 - x - x^2*y)) in the sense that T(n,k) is the coefficient of x^n*y^k in the expansion of the g.f.
T(n,0) = 1.
T(n,1) = n-2 for n>1.
T(n,2) = binomial(n-4,2) + n - 3 for n>3.
T(n,3) = binomial(n-6,3) + 2*binomial(n-5,2) for n>5.
T(4*m-3,2*m-2) = T(4*m-1,2*m-1) = m for m>0.
T(2*n+1,n-k) = A158909(n,k). (End)
T(n,k) = A348445(n-2,k) for n>1.

A348447 Irregular triangle read by rows: T(n,k) (n >= 0) is the number of ways of selecting k objects from n objects arranged in a circle with no two selected objects having unit separation (i.e. having exactly one object between them).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 1, 4, 4, 1, 5, 5, 1, 6, 9, 1, 7, 14, 7, 1, 8, 20, 16, 4, 1, 9, 27, 30, 9, 1, 10, 35, 50, 25, 1, 11, 44, 77, 55, 11, 1, 12, 54, 112, 105, 36, 4, 1, 13, 65, 156, 182, 91, 13, 1, 14, 77, 210, 294, 196, 49, 1, 15, 90, 275, 450, 378, 140, 15

Offset: 0

N. J. A. Sloane, Oct 23 2021

For odd n, T(n,k) is the number of independent vertex sets of size k in an n-cycle. For even n, T(n,k) is the number of independent vertex sets of size k in two disjoint (n/2)-cycles. Here, a cycle of length one or two is taken to be a path. - Andrew Howroyd, Jan 01 2024

			Triangle begins:
  1;
  1,   1;
  1,   2,   1;
  1,   3;
  1,   4,   4;
  1,   5,   5;
  1,   6,   9;
  1,   7,  14,   7;
  1,   8,  20,  16,   4;
  1,   9,  27,  30,   9;
  1,  10,  35,  50,  25;
  1,  11,  44,  77,  55,  11;
  1,  12,  54, 112, 105,  36,   4;
  1,  13,  65, 156, 182,  91,  13;
  ...
If n = 2 and we call the objects 0 and 1, the permitted sets of objects are {}, {0}, {1}, and {0,1}. If n = 3 and we call the objects 0, 1, and 2, then the permitted sets of objects are {}, {0}, {1}, and {2}; {0,1} is not a permitted set in this case since the objects lie in a circle and 2 lies between 0 and 1 in one direction. - _Michael A. Allen_, Apr 25 2022

Andrew Howroyd, Table of n, a(n) for n = 0..2578 (rows 0..100)
John Konvalina, On the number of combinations without unit separation., Journal of Combinatorial Theory, Series A 31.2 (1981): 101-107. See Table II (which erroneously lacks the n=k=2 element).

See A348445 for the case when the n objects are on a line.
The triangles A034807, A061896, A152060 are very similar.
The k=2 column is (essentially) A347553.

T(n) = {[Vecrev(p) | p <- Vec(-3 + y*x + y*(2 + y)*x^2 + (4 - 3*x - y*x^3)/((1 + y*x^2)*(1 - x - y*x^2)) + O(x*x^n))]}
{ my(A=T(10)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Jan 01 2024

G.f.: -3 + y*x + y*(2 + y)*x^2 + (4 - 3*x - y*x^3)/((1 + y*x^2)*(1 - x - y*x^2)). - Andrew Howroyd, Jan 01 2024

Corrected by Michael A. Allen, Apr 25 2022

Terms a(56) and beyond from Andrew Howroyd, Jan 01 2024

cp's OEIS Frontend

A335964 Triangle read by rows, T(n,k) = T(n-1,k) + T(n-3,k-1) + T(n-4,k-2) + delta(n,0)*delta(k,0), T(n,k<0) = T(n

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