A340660 A000079 is the first row. For the second row, subtract A001045. For the third row, subtract A001045 from the second one, etc. The resulting array is read by ascending antidiagonals.

1, 1, 2, 1, 1, 4, 1, 0, 3, 8, 1, -1, 2, 5, 16, 1, -2, 1, 2, 11, 32, 1, -3, 0, -1, 6, 21, 64, 1, -4, -1, -4, 1, 10, 43, 128, 1, -5, -2, -7, -4, -1, 22, 85, 256, 1, -6, -3, -10, -9, -12, 1, 42, 171, 512, 1, -7, -4, -13, -14, -23, -20, -1, 86, 341, 1024

Offset: 0

Author

Paul Curtz, Jan 15 2021

Keywords

• sign
• tabl
• easy

Comments

Every row has the signature (1,2).

(Among consequences: a(n) read by antidiagonals is

1,

1, 2,

1, 1, 4,

1, 0, 3, 8,

1, -1, 2, 5, 16

1, -2, 1, 2, 11, 32,

1, -3, 0, -1, 6, 21, 64,

... .

The row sums and their first two difference table terms are

1, 3, 6, 12, 23, 45, 88, ... = A086445(n+1) - 1

2, 3, 6, 11, 22, 43, 86, ... = A005578(n+2)

1, 3, 5, 11, 21, 43, 85, ... = A001045(n+2).

The antidiagonal sums are

b(n) = 1, 1, 3, 2, 5, 3, 9, 4, 15, 5, 27, 6, 49, 7, ... .)

Examples

			Square array:
1,  2,  4,   8,  16,  32,  64,  128, ... = A000079(n)
1,  1,  3,   5,  11,  21,  43,   85, ... = A001045(n+1)
1,  0,  2,   2,   6,  10,  22,   42, ... = A078008(n)
1, -1,  1,  -1,   1,  -1,   1,   -1, ... = A033999(n)
1, -2,  0,  -4,  -4, -12, -20,  -44, ... = -A084247(n)
1, -3, -1,  -7,  -9, -23, -41,  -87, ... = (-1)^n*A140966(n+1)
1, -4, -2, -10, -14, -34, -62, -130, ... = -A135440(n)
1, -5, -3, -13, -19, -45, -83, -173, ... = -A155980(n+3) or -A171382(n+1)
...
		

Crossrefs

Cf. A000079, A001045, A033999, A078008, A084247, A135440, A140966, A155980, A171382.

Cf. A005578, A086445.

Programs

Maple

A:= (n, k)-> (<<0|1>, <2|1>>^k. <<1, 2-n>>)[1$2]:
seq(seq(A(d-k, k), k=0..d), d=0..12);  # Alois P. Heinz, Jan 21 2021

Mathematica

A340660[m_, n_] := LinearRecurrence[{1, 2}, {1, m}, {n}]; Table[Reverse[Table[A340660[m, n + m - 2] // First, {m, 2, -n + 3, -1}]], {n, 1, 11}] // Flatten (* Robert P. P. McKone, Jan 28 2021 *)

PARI

T(n, k) = 2^k - n*(2^k - (-1)^k)/3;
matrix(10,10,n,k,T(n-1,k-1)) \\ Michel Marcus, Jan 19 2021

Formula

A(n,k) = 2^k - n*round(2^k/3).