A157522 Triangle T(n, k) = f(n, k) + f(n, n-k) - 1, where f(n, k) = k if k <= floor(n/4), floor(n/2) - k if floor(n/4) < k <= floor(n/2), k - floor(n/2) if floor(n/2) < k <= floor(3*n/4), otherwise n-k, read by rows.

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 1, 3, 1, 1, 3, 2, 2, 3, 1, 1, 3, 3, 1, 3, 3, 1, 1, 3, 4, 2, 2, 4, 3, 1, 1, 3, 5, 3, 1, 3, 5, 3, 1, 1, 3, 5, 4, 2, 2, 4, 5, 3, 1, 1, 3, 5, 5, 3, 1, 3, 5, 5, 3, 1, 1, 3, 5, 6, 4, 2, 2, 4, 6, 5, 3, 1, 1, 3, 5, 7, 5, 3, 1, 3, 5, 7, 5, 3, 1, 1, 3, 5, 7, 6, 4, 2, 2, 4, 6, 7, 5, 3, 1

Offset: 0

Author

Roger L. Bagula, Mar 02 2009

Keywords

• nonn
• tabl

Examples

			Triangle begins as:
  1;
  1, 1;
  1, 1, 1;
  1, 2, 2, 1;
  1, 3, 1, 3, 1;
  1, 3, 2, 2, 3, 1;
  1, 3, 3, 1, 3, 3, 1;
  1, 3, 4, 2, 2, 4, 3, 1;
  1, 3, 5, 3, 1, 3, 5, 3, 1;
  1, 3, 5, 4, 2, 2, 4, 5, 3, 1;
  1, 3, 5, 5, 3, 1, 3, 5, 5, 3, 1;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A157523.

Programs

Mathematica

f[n_, k_]= 1 +If[k<=Floor[n/4], k, If[Floor[n/4]G. C. Greubel, Jan 22 2022 *)

Sage

def f(n, k):
    if (k <= (n//4)): return k+1
    elif ((n//4) < k <= (n//2)): return (n//2)-k+1
    elif ((n//2) < k <= (3*n//4)): return k+1-(n//2)
    else: return n-k+1
def T(n,k): return f(n,k) + f(n,n-k) - 1
flatten([[T(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Jan 22 2022

Formula

T(n, k) = f(n, k) + f(n, n-k) - 1, where f(n, k) = k if k <= floor(n/4), floor(n/2) - k if floor(n/4) < k <= floor(n/2), k - floor(n/2) if floor(n/2) < k <= floor(3*n/4), otherwise n-k.

From G. C. Greubel, Jan 22 2022: (Start)

T(n, n-k) = T(n, k).

T(2*n, n) = 1.

T(2*n+1, n) = A040000(n).

Sum_{k=0..n} T(n, k) = A302488(n). (End)

Extensions

Edited by N. J. A. Sloane, Mar 05 2009