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COMMENTS
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Difference table D(n,k) = D(n-1,k+1) - D(n-1,k), D(0,k) = a(k):
4, -3, 5, -1, 9, 7, 25, ...
-7, 8, -6, 10, -2, 18, 14, 50, ...
15, -14, 16, -12, 20, -4, 36, 28, 100, ...
-29, 30, -28, 32, -24, 40, -8, 72, 56, 200, ...
59, -58, 60, -56, 64, -48, 80, -16, 144, 112, 400, ...
...
The diagonals are given by D(n,n+k) = a(k)*2^n.
a(n) - a(n) = 0, 0, 0, 0, 0, ... (trivially)
a(n+1) + a(n) = 1, 2, 4, 8, 16, ... = 2^n (by definition)
a(n+2) - a(n) = 1, 2, 4, 8, 16, ... = 2^n
a(n+3) + a(n) = 3, 6, 12, 24, 48, ... = 2^n*3
a(n+4) - a(n) = 5, 10, 20, 40, 80, ... = 2^n*5
a(n+5) + a(n) = 11, 22, 44, 88, 176, ... = 2^n*11
(...)
This table is given by T(r,n) = A001045(r)*2^n with r, n >= 0.
Sums of antidiagonals are A045883(n).
First upper diagonal: A054881(n+1).
Now consider the array from c(n) = (-1)^n*a(n) with its difference table:
4, 3, 5, 1, 9, -7, 25, -39, ... = c(n)
-1, 2, -4, 8, -16, 32, -64, 128, ... = -A122803(n)
3, -6, 12, -24, 48, -96, 192, -384, ... =
-9, 18, -36, 72, -144, 288, -576, 1152, ...
27, -54, 108, -216, 432, -864, 1728, -3456, ...
...
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