A134231 Triangle T(n, k) = n -k +1 with T(n, n-1) = 2*n-1 and T(n, n) = 1, read by rows.

1, 3, 1, 3, 5, 1, 4, 3, 7, 1, 5, 4, 3, 9, 1, 6, 5, 4, 3, 11, 1, 7, 6, 5, 4, 3, 13, 1, 8, 7, 6, 5, 4, 3, 15, 1, 9, 8, 7, 6, 5, 4, 3, 17, 1, 10, 9, 8, 7, 6, 5, 4, 3, 19, 1, 11, 10, 9, 8, 7, 6, 5, 4, 3, 21, 1, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 23, 1, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 25, 1

Offset: 1

Author

Gary W. Adamson, Oct 14 2007

Keywords

• nonn
• tabl

Examples

			First few rows of the triangle are:
  1;
  3, 1;
  3, 5, 1;
  4, 3, 7, 1;
  5, 4, 3, 9,  1;
  6, 5, 4, 3, 11,  1;
  7, 6, 5, 4,  3, 13, 1;
  ...
		

Links

• G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Crossrefs

Cf. A004736, A134081, A134227.

Programs

Magma

A134231:= func< n,k | k eq n select 1 else k eq n-1 select 2*n-1 else n-k+1 >;
[A134231(n,k): k in [1..n], n in [1..15]]; // G. C. Greubel, Feb 17 2021

Mathematica

T[n_, k_]:= If[k==n, 1, If[k==n-1, 2*n-1, n-k+1]];
Table[T[n, k], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Feb 17 2021 *)

Sage

def A134231(n,k): return 1 if k==n else 2*n-1 if k==n-1 else n-k+1
flatten([[A134231(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Feb 17 2021

Formula

T(n, k) = A004736(n, k) + A134081(n, k) - I, an infinite lower triangular matrix, where I = Identity matrix.

From G. C. Greubel, Feb 17 2021: (Start)

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

Sum_{k=1..n} T(n, k) = (n-1)*(n+6)/2 + [n=1] = A134227(n). (End)

Extensions

More terms and title changed by G. C. Greubel, Feb 17 2021