A156663 Triangle by columns, powers of 2 interleaved with zeros.

1, 0, 1, 2, 0, 1, 0, 2, 0, 1, 4, 0, 2, 0, 1, 0, 4, 0, 2, 0, 1, 8, 0, 4, 0, 2, 0, 1, 0, 8, 0, 4, 0, 2, 0, 1, 16, 0, 8, 0, 4, 0, 2, 0, 1, 0, 16, 0, 8, 0, 4, 0, 2, 0, 1, 32, 0, 16, 0, 8, 0, 4, 0, 2, 0, 1, 0, 32, 0, 16, 0, 8, 0, 4, 0, 2, 0, 1

Offset

0,4

Comments

Eigensequence of the triangle = A001045.

Links

Table of n, a(n) for n=0..77.

D. E. Davenport, L. W. Shapiro and L. C. Woodson, The Double Riordan Group, The Electronic Journal of Combinatorics, 18(2) (2012).

Formula

Triangle by columns, (1, 0, 2, 0, 4, 0, 8, ...) in every column.

From Peter Bala, Aug 15 2021: (Start)

T(n,k) = sqrt(2)^((n - k)/2) * (1 + (-1)^(n-k))/2 for 0 <= k <= n.

Double Riordan array (1/(1 - 2*x^2); x, x) as defined in Davenport et al.

The m-th power of the array is the double Riordan array (1/(1 - 2*x^2)^(m); x, x). Cf. A158944. (End)

Example

First few rows of the triangle =
   1;
   0,  1;
   2,  0,  1;
   0,  2,  0,  1;
   4,  0,  2,  0,  1;
   0,  4,  0,  2,  0,  1;
   8,  0,  4,  0,  2,  0,  1;
   0,  8,  0,  4,  0,  2,  0,  1;
  16,  0,  8,  0,  4,  0,  2,  0,  1;
   0, 16,  0,  8,  0,  4,  0,  2,  0,  1;
  32,  0, 16,  0,  8,  0,  4,  0,  2,  0,  1;
   0, 32,  0, 16,  0,  8,  0,  4,  0,  2,  0,  1;
  ...
The inverse array begins
   1;
   0,  1;
  -2,  0,  1;
   0, -2,  0,  1;
   0,  0, -2,  0,  1;
   0,  0,  0, -2,  0,  1;
   0,  0,  0,  0, -2,  0,  1;
   0,  0,  0,  0,  0, -2,  0,  1;
   0,  0,  0,  0,  0,  0, -2,  0,  1;
   ... - Peter Bala, Aug 15 2021

Maple

seq(seq( sqrt(2)^(n-k) * (1 + (-1)^(n-k))/2, k = 0..n), n = 0..10) # Peter Bala, Aug 15 2021

Crossrefs

Cf. A001045, A158944.

Sequence in context: A049771 A352514 A158944 * A139366 A049767 A286351

Adjacent sequences: A156660 A156661 A156662 * A156664 A156665 A156666

Keyword

nonn,tabl

Author

Gary W. Adamson, Feb 12 2009

Extensions

Typo in Data corrected by Peter Bala, Aug 15 2021

Status

approved