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

1,3

Comments

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

Links

Table of n, a(n) for n=1..60.

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)

Example

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

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 *)

Crossrefs

Cf. A204201.

Sequence in context: A193586 A007397 A362489 * A261721 A275490 A052345

Adjacent sequences: A204200 A204201 A204202 * A204204 A204205 A204206

Keyword

nonn,tabl

Author

Clark Kimberling, Jan 12 2012

Status

approved