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

0,8

Links

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

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)

Example

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;

Mathematica

f[n_, k_]= 1 +If[k<=Floor[n/4], k, If[Floor[n/4]<k<=Floor[n/2], Floor[n/2]-k, If[Floor[n/2]<k<=Floor[3*n/4], k-Floor[n/2], n-k]]];
T[n_, k_]:= f[n, k] +f[n, n-k] -1;
Table[T[n, k], {n, 0, 15}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Jan 22 2022 *)

Prog

(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

Crossrefs

Cf. A157523.

Sequence in context: A353468 A184305 A337279 * A059674 A367624 A342748

Adjacent sequences: A157519 A157520 A157521 * A157523 A157524 A157525

Keyword

nonn,tabl

Author

Roger L. Bagula, Mar 02 2009

Extensions

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

Status

approved