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

1,1

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

Links

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

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)

Example

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;
  ...

Mathematica

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

Crossrefs

Cf. A087323, A099035, A190730.

Sequence in context: A073653 A225487 A145398 * A094747 A300783 A359115

Adjacent sequences: A087319 A087320 A087321 * A087323 A087324 A087325

Keyword

nonn,tabl,easy

Author

Amarnath Murthy, Sep 03 2003

Extensions

Edited and extended by David Wasserman, May 06 2005

Name edited by Petros Hadjicostas, Feb 15 2021

Status

approved