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A319047 Square array A(n,k) where column k is balanced (2k+1)-ary enumeration of integers; n>=0, k>=1, read by antidiagonals.

%I A319047 #50 Feb 09 2021 06:50:51
%S A319047 0,0,1,0,1,-1,0,1,2,3,0,1,2,-2,4,0,1,2,3,-1,2,0,1,2,3,-3,5,-3,0,1,2,3,
%T A319047 4,-2,6,-2,0,1,2,3,4,-4,-1,7,-4,0,1,2,3,4,5,-3,7,3,9,0,1,2,3,4,5,-5,
%U A319047 -2,8,4,10,0,1,2,3,4,5,6,-4,-1,9,10,8,0,1,2,3,4,5,6,-6,-3,9,10,11,12
%N A319047 Square array A(n,k) where column k is balanced (2k+1)-ary enumeration of integers; n>=0, k>=1, read by antidiagonals.
%H A319047 Alois P. Heinz, <a href="/A319047/b319047.txt">Antidiagonals n = 0..200, flattened</a>
%H A319047 Wikipedia, <a href="https://en.wikipedia.org/wiki/Balanced_ternary">Balanced ternary</a>
%e A319047 Square array A(n,k) begins:
%e A319047    0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0, ...
%e A319047    1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
%e A319047   -1,  2,  2,  2,  2,  2,  2,  2,  2,  2,  2, ...
%e A319047    3, -2,  3,  3,  3,  3,  3,  3,  3,  3,  3, ...
%e A319047    4, -1, -3,  4,  4,  4,  4,  4,  4,  4,  4, ...
%e A319047    2,  5, -2, -4,  5,  5,  5,  5,  5,  5,  5, ...
%e A319047   -3,  6, -1, -3, -5,  6,  6,  6,  6,  6,  6, ...
%e A319047   -2,  7,  7, -2, -4, -6,  7,  7,  7,  7,  7, ...
%e A319047   -4,  3,  8, -1, -3, -5, -7,  8,  8,  8,  8, ...
%e A319047    9,  4,  9,  9, -2, -4, -6, -8,  9,  9,  9, ...
%e A319047   10, 10, 10, 10, -1, -3, -5, -7, -9, 10, 10, ...
%p A319047 A:= proc(n, k) option remember; `if`(n=0, 0,
%p A319047       (b-> b*A(iquo(n, b), k)+mods(n, b))(2*k+1))
%p A319047     end:
%p A319047 seq(seq(A(n, 1+d-n), n=0..d), d=0..14);
%t A319047 A[n_, k_] := A[n, k] = If[n == 0, 0, With[{b = 2k+1},
%t A319047      b*A[Quotient[n, b], k] + Mod[n, b, -Quotient[b-1, 2]]]];
%t A319047 Table[Table[A[n, 1+d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* _Jean-François Alcover_, Feb 09 2021, after _Alois P. Heinz_ *)
%Y A319047 Columns k=1-4 give: A117966, A309991, A309995, A316823.
%Y A319047 A(n,n+1) gives A001477.
%Y A319047 A(n+1,n) gives A001478 (for n>0).
%K A319047 sign,look,hear,tabl
%O A319047 0,9
%A A319047 _Alois P. Heinz_, Aug 26 2019