

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.


5



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
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OFFSET

0,7


COMMENTS

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.


LINKS

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


FORMULA

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


EXAMPLE

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.


PROG

(Scheme)
(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)))))))))


CROSSREFS

The topmost row and the leftmost column: A034968.
Only the lower triangular region: A231714. Related tables: A230415, A230419. Cf. also A101080, A231717.
Sequence in context: A268038 A274923 A249071 * A340945 A224898 A027360
Adjacent sequences: A231710 A231711 A231712 * A231714 A231715 A231716


KEYWORD

nonn,base,tabl


AUTHOR

Antti Karttunen, Nov 12 2013


STATUS

approved



