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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.

%I A340660 #31 Jan 30 2021 11:50:20
%S A340660 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,
%T A340660 -4,-1,-4,1,10,43,128,1,-5,-2,-7,-4,-1,22,85,256,1,-6,-3,-10,-9,-12,1,
%U A340660 42,171,512,1,-7,-4,-13,-14,-23,-20,-1,86,341,1024
%N 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.
%C A340660 Every row has the signature (1,2).
%C A340660 (Among consequences: a(n) read by antidiagonals is
%C A340660 1,
%C A340660 1,  2,
%C A340660 1,  1, 4,
%C A340660 1,  0, 3,  8,
%C A340660 1, -1, 2,  5, 16
%C A340660 1, -2, 1,  2, 11, 32,
%C A340660 1, -3, 0, -1,  6, 21, 64,
%C A340660 ... .
%C A340660 The row sums and their first two difference table terms are
%C A340660 1, 3, 6, 12, 23, 45, 88, ... = A086445(n+1) - 1
%C A340660 2, 3, 6, 11, 22, 43, 86, ... = A005578(n+2)
%C A340660 1, 3, 5, 11, 21, 43, 85, ... = A001045(n+2).
%C A340660 The antidiagonal sums are
%C A340660 b(n) = 1, 1, 3, 2, 5, 3, 9, 4, 15, 5, 27, 6, 49, 7, ... .)
%F A340660 A(n,k) = 2^k - n*round(2^k/3).
%e A340660 Square array:
%e A340660 1,  2,  4,   8,  16,  32,  64,  128, ... = A000079(n)
%e A340660 1,  1,  3,   5,  11,  21,  43,   85, ... = A001045(n+1)
%e A340660 1,  0,  2,   2,   6,  10,  22,   42, ... = A078008(n)
%e A340660 1, -1,  1,  -1,   1,  -1,   1,   -1, ... = A033999(n)
%e A340660 1, -2,  0,  -4,  -4, -12, -20,  -44, ... = -A084247(n)
%e A340660 1, -3, -1,  -7,  -9, -23, -41,  -87, ... = (-1)^n*A140966(n+1)
%e A340660 1, -4, -2, -10, -14, -34, -62, -130, ... = -A135440(n)
%e A340660 1, -5, -3, -13, -19, -45, -83, -173, ... = -A155980(n+3) or -A171382(n+1)
%e A340660 ...
%p A340660 A:= (n, k)-> (<<0|1>, <2|1>>^k. <<1, 2-n>>)[1$2]:
%p A340660 seq(seq(A(d-k, k), k=0..d), d=0..12);  # _Alois P. Heinz_, Jan 21 2021
%t A340660 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 *)
%o A340660 (PARI) T(n, k) = 2^k - n*(2^k - (-1)^k)/3;
%o A340660 matrix(10,10,n,k,T(n-1,k-1)) \\ _Michel Marcus_, Jan 19 2021
%Y A340660 Cf. A000079, A001045, A033999, A078008, A084247, A135440, A140966, A155980, A171382.
%Y A340660 Cf. A005578, A086445.
%K A340660 sign,tabl,easy
%O A340660 0,3
%A A340660 _Paul Curtz_, Jan 15 2021