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

1,2

Links

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

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)

Example

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;
  ...

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 *)

Prog

(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
(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

Crossrefs

Cf. A004736, A134081, A134227.

Sequence in context: A210952 A208523 A209572 * A225598 A126637 A110091

Adjacent sequences: A134228 A134229 A134230 * A134232 A134233 A134234

Keyword

nonn,tabl

Author

Gary W. Adamson, Oct 14 2007

Extensions

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

Status

approved