A204207 Triangle based on (1,2,3) averaging array.

2, 3, 5, 5, 8, 11, 9, 13, 19, 23, 17, 22, 32, 42, 47, 33, 39, 54, 74, 89, 95, 65, 72, 93, 128, 163, 184, 191, 129, 137, 165, 221, 291, 347, 375, 383, 257, 266, 302, 386, 512, 638, 722, 758, 767, 513, 523, 568, 688, 898, 1150, 1360, 1480, 1525, 1535, 1025

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:
2
3....5
5....8....11
9....13...19...23
17...22...32...42...47
33...39...54...74...89...95
		

Crossrefs

Cf. A204201.

Programs

Mathematica

a = 1; r = 2; b = 3;
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[2 (1/2) (1/r) 2^n*u[n], {n, 1, 12}];
TableForm[u] (* A204207 triangle *)
Flatten[u]   (* A204207 sequence *)

Formula

T(n,n) = A083329(n). - Philippe Deléham, Dec 24 2013

T(n,1) = A000051(n-1). - Philippe Deléham, Dec 24 2013

Sum_{k=1..n} T(n,k)=A036289(n). - Philippe Deléham, Dec 24 2013

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)=3, T(2,2)=5, T(n,k)=0 if k<1 or if k>n. - Philippe Deléham, Dec 24 2013

Extensions

Example corrected by Philippe Deléham, Dec 22 2013