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

0,3

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, ... .)

Links

Table of n, a(n) for n=0..65.

Formula

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

Example

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)
...

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 *)

Prog

(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

Crossrefs

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

Cf. A005578, A086445.

Sequence in context: A251636 A248939 A106246 * A136674 A144383 A205553

Adjacent sequences: A340657 A340658 A340659 * A340661 A340662 A340663

Keyword

sign,tabl,easy

Author

Paul Curtz, Jan 15 2021

Status

approved