A157458 Triangle, read by rows, double tent function: T(n,k) = min(1 + 2*k, 1 + 2*(n-k), n).

0, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 3, 4, 3, 1, 1, 3, 5, 5, 3, 1, 1, 3, 5, 6, 5, 3, 1, 1, 3, 5, 7, 7, 5, 3, 1, 1, 3, 5, 7, 8, 7, 5, 3, 1, 1, 3, 5, 7, 9, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 10, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 11, 11, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 11, 12, 11, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 11, 13, 13, 11, 9, 7, 5, 3, 1

Offset: 0

Author

Roger L. Bagula, Mar 01 2009

Keywords

• nonn
• tabl
• easy

Comments

The general form of this, and related triangular sequences, takes the form A(n, k, m) = (m*(n-k) + 1)*A(n-1, k-1, m) + (m*k + 1)*A(n-1, k, m) + m*f(n, k)* A(n-2, k-1, m), where f(n,k) is a polynomial in n and k.

Row sums are: 0, 2, 4, 8, 12, 18, 24, 32, 40, 50, 60, ... = A007590(n+1). - N. J. A. Sloane, Aug 27 2009

Examples

			Triangle begins as:
  0;
  1, 1;
  1, 2, 1;
  1, 3, 3, 1;
  1, 3, 4, 3, 1;
  1, 3, 5, 5, 3, 1;
  1, 3, 5, 6, 5, 3, 1;
  1, 3, 5, 7, 7, 5, 3, 1;
  1, 3, 5, 7, 8, 7, 5, 3, 1;
  1, 3, 5, 7, 9, 9, 7, 5, 3, 1;
  1, 3, 5, 7, 9, 10, 9, 7, 5, 3, 1;
		

Links

• Nathaniel Johnston, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A003983, A157457.

Programs

Maple

T := proc(m,n) return min(1+2*m, 1+2*(n-m), n): end: seq(seq(T(m,n),m=0..n),n=0..14); # Nathaniel Johnston, Apr 29 2011

Mathematica

T[n_, k_]:= Min[1+2*k, 1+2*(n-k), n]; Table[T[n, k], {n,0,14}, {k,0,n}]//Flatten

Formula

T(n, k) = min(1 + 2*k, 1 + 2*(n - k), n).

From Yu-Sheng Chang, May 19 2020: (Start)

O.g.f.: F(z,v) = (1+v)*z/((1-v*z-1)*(1-z)*(1-v*z^2)).

T(n,k) = [v^k] (1+v)*(2*v^(n+1)+2-((sqrt(v)-1)^2 * (-1)^n + (sqrt(v)+1)^2) * v^((1/2)*n))/(2*(v-1)^2). (End)

Extensions

Edited by N. J. A. Sloane, Aug 27 2009

More terms from and partially edited by G. C. Greubel, May 21 2020