A204202 Triangle based on (0,2/3,1) averaging array.

2, 2, 5, 2, 7, 11, 2, 9, 18, 23, 2, 11, 27, 41, 47, 2, 13, 38, 68, 88, 95, 2, 15, 51, 106, 156, 183, 191, 2, 17, 66, 157, 262, 339, 374, 383, 2, 19, 83, 223, 419, 601, 713, 757, 767, 2, 21, 102, 306, 642, 1020, 1314, 1470, 1524, 1535, 2, 23, 123, 408, 948

Offset: 1

Author

Clark Kimberling, Jan 12 2012

Keywords

• nonn
• tabl

Comments

See A204201 for a discussion of averaging arrays and related triangles

Examples

			First six rows:
2
2...5
2...7....11
2...9....18...23
2...11...27...41...47
2...13...38...68...88..95
		

Crossrefs

Cf. A204201.

Programs

Mathematica

a = 0; r = 2/3; b = 1;
t[1, 1] = r;
t[n_, 1] := (a + t[n - 1, 1])/2;
t[n_, n_] := (b + t[n - 1, n - 1])/2;
t[n_, k_] := (t[n - 1, k - 1] + t[n - 1, k])/2;
u[n_] := Table[t[n, k], {k, 1, n}]
Table[u[n], {n, 1, 5}]   (* averaging array *)
u = Table[(1/r) 2^n*u[n], {n, 1, 12}];
TableForm[u]  (* A204202 triangle *)
Flatten[u]    (* A204202 sequence *)

Formula

From Philippe Deléham, Dec 24 2013: (Start)

T(n,n) = A055010(n) = A083329(n) = A153893(n-1).

Sum_{k=1..n} T(n,k) = A066373(n+1).

T(n,k) = T(n-1,k)+3*T(n-1,k-1)-2*T(n-2,k-1)-2*T(n-2,k-2), T(1,1)=2, T(2,1)=2, T(2,2)=5, T(n,k)=0 if k<1 or if k>n. (End)