A204203 Triangle based on (0,1/4,1) averaging array.

1, 1, 5, 1, 6, 13, 1, 7, 19, 29, 1, 8, 26, 48, 61, 1, 9, 34, 74, 109, 125, 1, 10, 43, 108, 183, 234, 253, 1, 11, 53, 151, 291, 417, 487, 509, 1, 12, 64, 204, 442, 708, 904, 996, 1021, 1, 13, 76, 268, 646, 1150, 1612, 1900, 2017, 2045, 1, 14, 89, 344, 914

Offset: 1

Author

Clark Kimberling, Jan 12 2012

Keywords

• nonn
• tabl

Comments

See A204201 for a discussion and guide to other averaging arrays.

Examples

			First six rows:
1
1...5
1...6...13
1...7...19...29
1...8...26...48...61
1...9...34...74...109...125
		

Crossrefs

Cf. A204201.

Programs

Mathematica

a = 0; r = 1/4; 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[(1/2) (1/r) 2^n*u[n], {n, 1, 12}];
TableForm[u]  (* A204203 triangle *)
Flatten[u]    (* A204203 sequence *)

Formula

From Philippe Deléham, Dec 24 2013: (Start)

T(n,n) = A036563(n+1).

Sum_{k=1..n} T(n,k) = A014480(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)=1, T(2,1)=1, T(2,2)=5, T(n,k)=0 if k<1 or if k>n. (End)