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

1,2

Links

Table of n, a(n) for n=1..105.

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)

Example

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;
  ...

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 *)

Prog

(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

Crossrefs

Row sums: A016777.

Cf. A130296, A051340, A016777, A131830.

Sequence in context: A162911 A245327 A352680 * A360913 A204123 A237448

Adjacent sequences: A131818 A131819 A131820 * A131822 A131823 A131824

Keyword

nonn,tabl,easy

Author

Gary W. Adamson, Jul 19 2007

Extensions

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

Status

approved