A231714 -id:A231714 - cp's OEIS frontend

A230417 Lower triangular region of A230415, a triangular table read by rows: T(n, k) tells in how many digit positions the factorial base representations (A007623) of n and k differ, where (n, k) = (0,0), (1,0), (1,1), (2,0), (2,1), (2,2), ..., n >= 0 and (0 <= k <= n).

0, 1, 0, 1, 2, 0, 2, 1, 1, 0, 1, 2, 1, 2, 0, 2, 1, 2, 1, 1, 0, 1, 2, 2, 3, 2, 3, 0, 2, 1, 3, 2, 3, 2, 1, 0, 2, 3, 1, 2, 2, 3, 1, 2, 0, 3, 2, 2, 1, 3, 2, 2, 1, 1, 0, 2, 3, 2, 3, 1, 2, 1, 2, 1, 2, 0, 3, 2, 3, 2, 2, 1, 2, 1, 2, 1, 1, 0, 1, 2, 2, 3, 2, 3, 1, 2, 2, 3, 2, 3, 0, 2, 1, 3, 2, 3, 2, 2, 1, 3, 2, 3, 2, 1, 0, 2, 3, 1, 2, 2, 3, 2, 3, 1, 2, 2, 3, 1, 2, 0

Offset: 0

Antti Karttunen, Nov 10 2013

			This triangular table begins:
  0;
  1, 0;
  1, 2, 0;
  2, 1, 1, 0;
  1, 2, 1, 2, 0;
  2, 1, 2, 1, 1, 0;
  1, 2, 2, 3, 2, 3, 0;
  ...
Please see A230415 for examples showing how the terms are computed.

Antti Karttunen, Rows n = 0..120 of triangle, flattened

This is a lower, or equivalently, an upper triangular subregion of symmetric square array A230415.

Cf. A231714, A060130, A055881.

(define (A230417 n) (A230415bi (A003056 n) (A002262 n)))
(define (A230415bi x y) (let loop ((x x) (y y) (i 2) (d 0)) (cond ((and (zero? x) (zero? y)) d) (else (loop (floor->exact (/ x i)) (floor->exact (/ y i)) (+ i 1) (+ d (if (= (modulo x i) (modulo y i)) 0 1)))))))

a(n) = A230415bi(A003056(n),A002262(n)). [As a sequence, this is obtained by taking a subsection from array A230415.]
T(n,0) = A060130(n) [the leftmost column].
For n >= 1, T(n,n-1) = A055881(n) [the last nonzero column].
Each entry T(n,k) <= A231714(n,k).

A231713 Square array A(i,j) = the sum of absolute values of digit differences in the matching positions of the factorial base representations of i and j, for i >= 0, j >= 0, read by antidiagonals.

Original entry on oeis.org

0, 1, 1, 1, 0, 1, 2, 2, 2, 2, 2, 1, 0, 1, 2, 3, 3, 1, 1, 3, 3, 1, 2, 1, 0, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 1, 0, 1, 2, 1, 2, 3, 3, 3, 3, 1, 1, 3, 3, 3, 3, 3, 2, 1, 2, 3, 0, 3, 2, 1, 2, 3, 4, 4, 2, 2, 4, 4, 4, 4, 2, 2, 4, 4, 2, 3, 2, 1, 2, 3, 0, 3, 2, 1, 2, 3, 2, 3, 3, 3, 3, 3, 3, 1, 1, 3, 3, 3, 3, 3, 3, 3, 2, 3, 2, 1, 2, 1, 0, 1, 2, 1, 2, 3, 2, 3

Offset: 0

Antti Karttunen, Nov 12 2013

This table relates to the factorial base representation (A007623) in a similar way as A101080 relates to the binary system. See A230415 for another analog.

			The top left corner of this square array begins as:
0, 1, 1, 2, 2, 3, 1, 2, 2, 3, 3, ...
1, 0, 2, 1, 3, 2, 2, 1, 3, 2, 4, ...
1, 2, 0, 1, 1, 2, 2, 3, 1, 2, 2, ...
2, 1, 1, 0, 2, 1, 3, 2, 2, 1, 3, ...
2, 3, 1, 2, 0, 1, 3, 4, 2, 3, 1, ...
3, 2, 2, 1, 1, 0, 4, 3, 3, 2, 2, ...
1, 2, 2, 3, 3, 4, 0, 1, 1, 2, 2, ...
2, 1, 3, 2, 4, 3, 1, 0, 2, 1, 3, ...
2, 3, 1, 2, 2, 3, 1, 2, 0, 1, 1, ...
3, 2, 2, 1, 3, 2, 2, 1, 1, 0, 2, ...
3, 4, 2, 3, 1, 2, 2, 3, 1, 2, 0, ...
...
For example, A(1,2) = A(2,1) = 2 as 1 has factorial base representation '...0001' and 2 has factorial base representation '...0010', and adding the absolute values of the digit differences, we get 1+1 = 2.
On the other hand, A(3,5) = A(5,3) = 1, as 3 has factorial base representation '...0011' and 5 has factorial base representation '...0021', and they differ only by their second rightmost digit, the absolute value of difference being 1.
Note that as A007623(6)='100' and A007623(10)='120', we have A(6,10) = A(10,6) = 2.

Antti Karttunen, The first 121 antidiagonals of the table, flattened

The topmost row and the leftmost column: A034968.

Only the lower triangular region: A231714. Related tables: A230415, A230419. Cf. also A101080, A231717.

(define (A231713 n) (A231713bi (A025581 n) (A002262 n)))
(define (A231713bi x y) (let loop ((x x) (y y) (i 2) (d 0)) (cond ((and (zero? x) (zero? y)) d) (else (loop (floor->exact (/ x i)) (floor->exact (/ y i)) (+ i 1) (+ d (abs (- (modulo x i) (modulo y i)))))))))

Each entry A(i,j) >= A230415(i,j) and also each entry A(i,j) >= abs(A230419(i,j)).

cp's OEIS Frontend

A230417 Lower triangular region of A230415, a triangular table read by rows: T(n, k) tells in how many digit positions the factorial base representations (A007623) of n and k differ, where (n, k) = (0,0), (1,0), (1,1), (2,0), (2,1), (2,2), ..., n >= 0 and (0 <= k <= n).

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