A131821 Triangle read by rows: row n consists of n followed by (n-2) ones then n.

1, 2, 2, 3, 1, 3, 4, 1, 1, 4, 5, 1, 1, 1, 5, 6, 1, 1, 1, 1, 6, 7, 1, 1, 1, 1, 1, 7, 8, 1, 1, 1, 1, 1, 1, 8, 9, 1, 1, 1, 1, 1, 1, 1, 9, 10, 1, 1, 1, 1, 1, 1, 1, 1, 10, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 12, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 14, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 14

Offset: 1

Author

Gary W. Adamson, Jul 19 2007

Keywords

• nonn
• tabl
• easy

Examples

			First few rows of the triangle are:
  1;
  2, 2;
  3, 1, 3;
  4, 1, 1, 4;
  5, 1, 1, 1, 5;
  6, 1, 1, 1, 1, 6;
  7, 1, 1, 1, 1, 1, 7;
  ...
		

Crossrefs

Row sums: A016777.

Cf. A130296, A051340, A016777, A131830.

Programs

Maple

A131821 := proc(n,c) if c=1 or c=n then n ; else 1 ; fi ; end: for n from 1 to 16 do for c from 1 to n do printf("%d,",A131821(n,c)) ; od: od: # R. J. Mathar, May 30 2008

Mathematica

T[n_, k_] := If[k == 1 || k == n, n, 1];
Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, May 05 2023 *)

Maxima

T(n,k) := if k = 1 or k = n then n else 1$
create_list(T(n, k), n, 1, 12, k, 1, n); /* Franck Maminirina Ramaharo, Dec 19 2018 */

Python

from math import isqrt
def A131821(n): return m+(k>r) if 0<=(k:=n<<1)-(r:=(m:=isqrt(k))*(m+1))<=2 or n<=2 else 1 # Chai Wah Wu, Nov 07 2024

Formula

T(n,k) = A130296(n,k) + A051340(k-1,n-k) - 1.

T(n,1) = T(n,n) = n for n >= 1, and T(n,k) = 1 for 2 <= k <= n - 1, n >= 3.

From Franck Maminirina Ramaharo, Dec 19 2018: (Start)

T(n,k) = A130296(n,k) + A130296(n,n-k+1) - 1.

G.f.: y*(x - 3*x^2*y^2 + (x^2 + x^3)*y^3)/((1 - y)^2*(1 - x*y)^2).

E.g.f.: (1 - x^2 +(x^2 + (x - x^2)*y)*exp(y) - (1 - (x - x^2)*y)*exp(x*y))/(1 - x). (End)

Extensions

More terms from R. J. Mathar, May 30 2008