A309991 -id:A309991 - cp's OEIS frontend

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

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, 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, -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

Offset: 0

Alois P. Heinz, Aug 26 2019

			Square array A(n,k) begins:
   0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0, ...
   1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
  -1,  2,  2,  2,  2,  2,  2,  2,  2,  2,  2, ...
   3, -2,  3,  3,  3,  3,  3,  3,  3,  3,  3, ...
   4, -1, -3,  4,  4,  4,  4,  4,  4,  4,  4, ...
   2,  5, -2, -4,  5,  5,  5,  5,  5,  5,  5, ...
  -3,  6, -1, -3, -5,  6,  6,  6,  6,  6,  6, ...
  -2,  7,  7, -2, -4, -6,  7,  7,  7,  7,  7, ...
  -4,  3,  8, -1, -3, -5, -7,  8,  8,  8,  8, ...
   9,  4,  9,  9, -2, -4, -6, -8,  9,  9,  9, ...
  10, 10, 10, 10, -1, -3, -5, -7, -9, 10, 10, ...

Alois P. Heinz, Antidiagonals n = 0..200, flattened
Wikipedia, Balanced ternary

Columns k=1-4 give: A117966, A309991, A309995, A316823.
A(n,n+1) gives A001477.
A(n+1,n) gives A001478 (for n>0).

A:= proc(n, k) option remember; `if`(n=0, 0,
      (b-> b*A(iquo(n, b), k)+mods(n, b))(2*k+1))
    end:
seq(seq(A(n, 1+d-n), n=0..d), d=0..14);

A[n_, k_] := A[n, k] = If[n == 0, 0, With[{b = 2k+1},
     b*A[Quotient[n, b], k] + Mod[n, b, -Quotient[b-1, 2]]]];
Table[Table[A[n, 1+d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Feb 09 2021, after Alois P. Heinz *)

A309995 Balanced septenary enumeration (or balanced septenary representation) of integers; write n in septenary and then replace 4's with (-3),s, 5's with (-2)'s, and 6's with (-1)'s.

Original entry on oeis.org

0, 1, 2, 3, -3, -2, -1, 7, 8, 9, 10, 4, 5, 6, 14, 15, 16, 17, 11, 12, 13, 21, 22, 23, 24, 18, 19, 20, -21, -20, -19, -18, -24, -23, -22, -14, -13, -12, -11, -17, -16, -15, -7, -6, -5, -4, -10, -9, -8, 49, 50, 51, 52, 46, 47, 48, 56, 57, 58, 59, 53, 54, 55, 63

Offset: 0

Jackson Haselhorst, Aug 26 2019

This sequence, like the balanced ternary and quinary sequences, includes every integer exactly once.

			As 54_10 = 105_7, the digits of 54 in base 7 are 1, 0 and 5. 5 > 3 so it's replaced by -2. The digits then are 1, 0 and -2 giving a(54) = 1*7^2 + 0 * 7^1 + (-2) * 7^0 = 49 + 0 - 2 = 47. - _David A. Corneth_, Aug 26 2019

Alois P. Heinz, Table of n, a(n) for n = 0..16806

Cf. A007093, A117966, A309991.

Column k=3 of A319047.

a:= proc(n) option remember; `if`(n=0, 0,
      7*a(iquo(n, 7))+mods(n, 7))
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Aug 26 2019

a(n,b=7) = fromdigits(apply(d -> if (dRémy Sigrist, Aug 26 2019

PARI

a(n) = my(d = digits(n, 7)); for(i = 1, #d, if(d[i] > 3, d[i]-=7)); fromdigits(d, 7) \\ David A. Corneth, Aug 26 2019

A316823 Balanced nonary enumeration (or balanced nonary representation) of integers; write n in nonary (base 9) and then replace 5's with (-4)'s, 6's with (-3)'s, 7's with (-2)'s, and 8's with (-1)'s.

Original entry on oeis.org

0, 1, 2, 3, 4, -4, -3, -2, -1, 9, 10, 11, 12, 13, 5, 6, 7, 8, 18, 19, 20, 21, 22, 14, 15, 16, 17, 27, 28, 29, 30, 31, 23, 24, 25, 26, 36, 37, 38, 39, 40, 32, 33, 34, 35, -36, -35, -34, -33, -32, -40, -39, -38, -37, -27, -26, -25, -24, -23, -31, -30, -29, -28

Offset: 0

Jackson Haselhorst, Aug 26 2019

			Since 35_10=38_9, the digits of 35 in base 9 are 3 and 8. 8>4, so it is replaced with (-1). The digits are then 3 and -1, so a(35)=3*9^1+(-1)*9^0=27-1=26.

Alois P. Heinz, Table of n, a(n) for n = 0..6560

Cf. A007095, A117966, A309991, A309995.

Column k=4 of A319047.

a:= proc(n) option remember; `if`(n=0, 0,
      9*a(iquo(n, 9))+mods(n, 9))
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Aug 26 2019

f(x) = if (x > 9/2, -(9-x), x);
a(n) = subst(Pol(apply(x->f(x), digits(n, 9)), 'x), 'x, 9); \\ Michel Marcus, Aug 27 2019

cp's OEIS Frontend

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

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A309995 Balanced septenary enumeration (or balanced septenary representation) of integers; write n in septenary and then replace 4's with (-3),s, 5's with (-2)'s, and 6's with (-1)'s.

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A316823 Balanced nonary enumeration (or balanced nonary representation) of integers; write n in nonary (base 9) and then replace 5's with (-4)'s, 6's with (-3)'s, 7's with (-2)'s, and 8's with (-1)'s.

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