A135087 Triangle T(n, k) = 2*A134058(n, k) - 1, read by rows.

1, 3, 3, 3, 7, 3, 3, 11, 11, 3, 3, 15, 23, 15, 3, 3, 19, 39, 39, 19, 3, 3, 23, 59, 79, 59, 23, 3, 3, 27, 83, 139, 139, 83, 27, 3, 3, 31, 111, 223, 279, 223, 111, 31, 3, 3, 35, 143, 335, 503, 503, 335, 143, 35, 3

Offset

0,2

Links

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

Formula

T(n, k) = 2*A134058(n, k) - 1.

From G. C. Greubel, May 03 2021: (Start)

T(n, k) = 4*binomial(n, k) - 2*[n=0] - 1.

Sum_{k=0..n} T(n, k) = 2^(n+2) - (n + 1 + 2*[n=0]) = A095768(n) - 2*[n=0]. (End)

Example

First few rows of the triangle are:
  1;
  3,  3;
  3,  7,  3;
  3, 11, 11,  3;
  3, 15, 23, 15,  3;
  3, 19, 39, 39, 19,  3;
  3, 23, 59, 79, 59, 23, 3;
  ...

Mathematica

Table[4*Binomial[n, k] -2*Boole[n==0] -1, {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, May 03 2021 *)

Prog

(Magma) [1] cat [4*Binomial(n, k) -1: k in [0..n], n in [1..12]]; // G. C. Greubel, May 03 2021
(Sage)
def A135087(n, k): return 4*binomial(n, k) -2*bool(n==0) -1
flatten([[A135087(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 03 2021

Crossrefs

Cf. A095768, A134058.

Sequence in context: A131757 A214834 A291767 * A294505 A242016 A084038

Adjacent sequences: A135084 A135085 A135086 * A135088 A135089 A135090

Keyword

nonn,tabl

Author

Gary W. Adamson, Nov 18 2007

Status

approved