A372727 Triangle read by rows: T(n, k) = n if k = 0, otherwise n - k*floor(n/k). The binary modulo operation.

0, 1, 0, 2, 0, 0, 3, 0, 1, 0, 4, 0, 0, 1, 0, 5, 0, 1, 2, 1, 0, 6, 0, 0, 0, 2, 1, 0, 7, 0, 1, 1, 3, 2, 1, 0, 8, 0, 0, 2, 0, 3, 2, 1, 0, 9, 0, 1, 0, 1, 4, 3, 2, 1, 0, 10, 0, 0, 1, 2, 0, 4, 3, 2, 1, 0, 11, 0, 1, 2, 3, 1, 5, 4, 3, 2, 1, 0, 12, 0, 0, 0, 0, 2, 0, 5, 4, 3, 2, 1, 0

Offset

0,4

Comments

This binary operation 'mod' and the convention to set x mod y = x if y = 0 is discussed in 'Concrete Mathematics' by Graham et. al. on p. 82 and the connection with the congruence relation '(mod)' on p. 123.

References

Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, 2nd ed., Addison-Wesley, 1994, 34th printing 2022.

Links

Table of n, a(n) for n=0..90.

Example

Triangle begins:
  [ 0]  0;
  [ 1]  1, 0;
  [ 2]  2, 0, 0;
  [ 3]  3, 0, 1, 0;
  [ 4]  4, 0, 0, 1, 0;
  [ 5]  5, 0, 1, 2, 1, 0;
  [ 6]  6, 0, 0, 0, 2, 1, 0;
  [ 7]  7, 0, 1, 1, 3, 2, 1, 0;
  [ 8]  8, 0, 0, 2, 0, 3, 2, 1, 0;
  [ 9]  9, 0, 1, 0, 1, 4, 3, 2, 1, 0;
  [10] 10, 0, 0, 1, 2, 0, 4, 3, 2, 1, 0;
  [11] 11, 0, 1, 2, 3, 1, 5, 4, 3, 2, 1, 0;
.
The triangle shows the modulo operation in the range 0 <= k <= n. Test your
computer implementation in the range R X R where R = [-6, ..., 0, ..., 6].
According to Graham et al. it should look like this:
   0, -1, -2,  0,  0, 0, -6, 0, 0, 0, 2, 4, 0
  -5,  0, -1, -2, -1, 0, -5, 0, 1, 1, 3, 0, 1
  -4, -4,  0, -1,  0, 0, -4, 0, 0, 2, 0, 1, 2
  -3, -3, -3,  0, -1, 0, -3, 0, 1, 0, 1, 2, 3
  -2, -2, -2, -2,  0, 0, -2, 0, 0, 1, 2, 3, 4
  -1, -1, -1, -1, -1, 0, -1, 0, 1, 2, 3, 4, 5
   0,  0,  0,  0,  0, 0,  0, 0, 0, 0, 0, 0, 0
  -5, -4, -3, -2, -1, 0,  1, 0, 1, 1, 1, 1, 1
  -4, -3, -2, -1,  0, 0,  2, 0, 0, 2, 2, 2, 2
  -3, -2, -1,  0, -1, 0,  3, 0, 1, 0, 3, 3, 3
  -2, -1,  0, -2,  0, 0,  4, 0, 0, 1, 0, 4, 4
  -1,  0, -3, -1, -1, 0,  5, 0, 1, 2, 1, 0, 5
   0, -4, -2,  0,  0, 0,  6, 0, 0, 0, 2, 1, 0

Maple

MOD := (n, k) -> ifelse(k = 0, n, n - k * iquo(n, k)):
seq( seq(MOD(n, k), k = 0..n), n = 0..12);

Prog

(Python)
def T(n, k): return n if k == 0 else n - k * (n // k)
for n in range(12): print([T(n, k) for k in range(n + 1)])
(Python)
def A372727_T(n, k): return n % k if k else n # Chai Wah Wu, May 14 2024

Crossrefs

Cf. A111490 (row sums).

Cf. A048158.

Sequence in context: A212163 A212195 A228926 * A321414 A268865 A024159

Adjacent sequences: A372724 A372725 A372726 * A372728 A372729 A372730

Keyword

nonn,tabl

Author

Peter Luschny, May 13 2024

Status

approved