A367824 Array read by ascending antidiagonals: A(n, k) is the numerator of (R(n) - k)/(n + k), where R(n) is the digit reversal of n, with A(0, 0) = 1.

1, 1, -1, 1, 0, -1, 1, 1, -1, -1, 1, 1, 0, -1, -1, 1, 3, 1, -1, -3, -1, 1, 2, 1, 0, -1, -2, -1, 1, 5, 3, 1, -1, -3, -5, -1, 1, 3, 1, 1, 0, -1, -1, -3, -1, 1, 7, 5, 1, 1, -1, -1, -5, -7, -1, 1, 4, 3, 2, 1, 0, -1, -2, -3, -4, -1, 1, 0, 7, 5, 3, 1, -1, -3, -5, -7, -9, -1

Offset

0,17

Comments

This array generalizes A367727.

Links

Stefano Spezia, First 151 antidiagonals of the array

Formula

A(1, n) = -A026741(n-1) for n > 0.

A(2, n) = -A060819(n-2) for n > 2.

A(3, n) = -A060789(n-3) for n > 3.

A(4, n) = -A106609(n-4) for n > 3.

A(5, n) = -A106611(n-5) for n > 4.

A(6, n) = -A051724(n-6) for n > 5.

A(7, n) = -A106615(n-7) for n > 6.

A(8, n) = -A106617(n-8) = A231190(n) for n > 7.

A(9, n) = -A106619(n-9) for n > 8.

A(10, n) = -A106612(n-10) for n > 9.

Example

The array of the fractions begins:
  1,  -1,   -1,   -1,   -1,   -1,    -1,    -1, ...
  1,   0, -1/3, -1/2, -3/5, -2/3,  -5/7,  -3/4, ...
  1, 1/3,    0, -1/5, -1/3, -3/7,  -1/2,  -5/9, ...
  1, 1/2,  1/5,    0, -1/7, -1/4,  -1/3,  -2/5, ...
  1, 3/5,  1/3,  1/7,    0, -1/9,  -1/5, -3/11, ...
  1, 2/3,  3/7,  1/4,  1/9,    0, -1/11,  -1/6, ...
  1, 5/7,  1/2,  1/3,  1/5, 1/11,     0, -1/13, ...
  1, 3/4,  5/9,  2/5, 3/11,  1/6,  1/13,     0, ...
  ...
The array of the numerators begins:
  1, -1, -1, -1, -1, -1, -1, -1, ...
  1,  0, -1, -1, -3, -2, -5, -3, ...
  1,  1,  0, -1, -1, -3, -1, -5, ...
  1,  1,  1,  0, -1, -1, -1, -2, ...
  1,  3,  1,  1,  0, -1, -1, -3, ...
  1,  2,  3,  1,  1,  0, -1, -1, ...
  1,  5,  1,  1,  1,  1,  0, -1, ...
  1,  3,  5,  2,  3,  1,  1,  0, ...
  ...

Mathematica

A[0, 0]=1; A[n_, k_]:=Numerator[(FromDigits[Reverse[IntegerDigits[n]]]-k)/(n+k)]; Table[A[n-k, k], {n, 0, 11}, {k, 0, n}]//Flatten

Crossrefs

Cf. A367825 (denominator), A367826 (antidiagonal sums).

Cf. A004086, A026741, A051724, A060789, A060819, A106609, A106611, A106612, A106617, A106619, A153881 (n=0), A231190, A367727 (k=1).

Sequence in context: A124921 A090623 A098094 * A087283 A355924 A111625

Adjacent sequences: A367821 A367822 A367823 * A367825 A367826 A367827

Keyword

sign,base,frac,look,tabl

Author

Stefano Spezia, Dec 02 2023

Status

approved