A157897 -id:A157897 - cp's OEIS frontend

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.

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

Michael A. Allen, Jul 30 2022

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.

			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;
...

Michael A. Allen, On a Two-Parameter Family of Generalizations of Pascal's Triangle, arXiv:2209.01377 [math.CO], 2022.
Michael A. Allen, On A Two-Parameter Family of Generalizations of Pascal's Triangle, J. Int. Seq. 25 (2022) Article 22.9.8.

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.

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}]

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.

A120415 Expansion of 1/(1-x-x^3-x^6).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 7, 11, 16, 25, 39, 59, 91, 141, 216, 332, 512, 787, 1210, 1863, 2866, 4408, 6783, 10436, 16054, 24700, 38002, 58464, 89947, 138385, 212903, 327550, 503937, 775304, 1192801, 1835123, 2823330, 4343681, 6682741, 10281375, 15817857, 24335721

Offset: 0

Jon E. Schoenfield, Aug 27 2006

Number of compositions of n into parts 1, 3, and 6. [Joerg Arndt, Sep 03 2013]

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
K. Edwards, A Pascal-like triangle related to the tribonacci numbers, Fib. Q., 46/47 (2008/2009), 18-25.
Index entries for linear recurrences with constant coefficients, signature (1,0,1,0,0,1).

Cf. A157897.

I:=[1,1,1,2,3,4]; [n le 6 select I[n] else Self(n-1)+Self(n-3)+Self(n-6): n in [1..40]]; // Vincenzo Librandi, Sep 03 2013

CoefficientList[Series[1/(1-x-x^3-x^6), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 03 2013 *)

def A120415_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1-x-x^3-x^6) ).list()
A120415_list(40) # G. C. Greubel, Sep 02 2022

a(n) = a(n-1) + a(n-3) + a(n-6).

a(n) = Sum_{k=0..floor(n/2)} A157897(n-k, k). - G. C. Greubel, Sep 02 2022

cp's OEIS Frontend

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.

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