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A156663
Triangle by columns, powers of 2 interleaved with zeros.
3
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
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
Sequence in context: A049771 A352514 A158944 * A139366 A049767 A286351
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Feb 12 2009
EXTENSIONS
Typo in Data corrected by Peter Bala, Aug 15 2021
STATUS
approved