A087322 Triangle T read by rows: T(n, 1) = 2*n + 1. For 1 < k <= n, T(n, k) = 2*T(n,k-1) + 1.

3, 5, 11, 7, 15, 31, 9, 19, 39, 79, 11, 23, 47, 95, 191, 13, 27, 55, 111, 223, 447, 15, 31, 63, 127, 255, 511, 1023, 17, 35, 71, 143, 287, 575, 1151, 2303, 19, 39, 79, 159, 319, 639, 1279, 2559, 5119, 21, 43, 87, 175, 351, 703, 1407, 2815, 5631, 11263, 23, 47, 95

Offset: 1

Author

Amarnath Murthy, Sep 03 2003

Keywords

• nonn
• tabl
• easy

Comments

With T(n,0) = n for n >= 0, this becomes J. M. Bergot's triangular array in the definition of A190730. - Petros Hadjicostas, Feb 15 2021

Examples

			Triangle T(n,k) (with rows n >= 1 and columns k = 1..n) begins:
   3;
   5, 11;
   7, 15, 31;
   9, 19, 39,  79;
  11, 23, 47,  95, 191;
  13, 27, 55, 111, 223, 447;
  15, 31, 63, 127, 255, 511, 1023;
  17, 35, 71, 143, 287, 575, 1151, 2303;
  19, 39, 79, 159, 319, 639, 1279, 2559, 5119;
  ...
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..11325 (rows 1..150 of the triangle, flattened)

Crossrefs

Cf. A087323, A099035, A190730.

Programs

Mathematica

A087322row[n_]:=NestList[2#+1&,2n+1,n-1];Array[A087322row,10] (* Paolo Xausa, Oct 17 2023 *)

Formula

T(n, k) = (n + 1)*2^k - 1 for n >= 1 and 1 <= k <= n.

From Petros Hadjicostas, Feb 15 2021: (Start)

Sum_{k=1..n} T(n,k) = A190730(n).

T(n,2) = 4*n + 3 for n >= 2.

T(n,n) = A087323(n).

T(n,n-1) = A099035(n) = (n+1)*2^(n-1) - 1 for n >= 2.

Recurrence: T(n,k) = 3*T(n,k-1) - 2*T(n,k-2) for n >= 2 and 2 <= k <= n with initial conditions the values of T(n, 1) and T(n,2).

Bivariate o.g.f.: Sum_{n,k>=1} T(n,k)*x^n*y^k = (4*x^3*y^2 - 2*x^2*y - 4*x*y - x + 3)*x*y/((1 - 2*x*y)^2*(1 - x*y)*(1 - x)^2). (End)

Extensions

Edited and extended by David Wasserman, May 06 2005

Name edited by Petros Hadjicostas, Feb 15 2021