A230419 Square array A(n,k) = difference of digit sums in factorial base representations (A007623) of n and k, n>=0, k>=0, read by antidiagonals; A(n,k) = A034968(n)-A034968(k).

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

Offset

0,7

Comments

Equivalently, A(n,k) = the sum of differences of digits in matching positions of the factorial base representations (A007623) of n and k.

Links

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

Formula

A(col,row) = A034968(col)-A034968(row). [Where col is the column and row the row index of entry A(col,row)]

Equally, as a sequence, a(n) = A034968(A025581(n)) - A034968(A002262(n)).

For each entry, A(j,i) = -A(i,j), or as a sequence, a(A061579(n)) = -a(n). [The array is symmetric up to the sign of entries]

Also, for each entry A(i,j), abs(A(i,j)) <= A231713(i,j).

Example

The top left corner array is:
   0,  1,  1,  2,  2,  3,  1,  2,  2,  3,  3, ...
  -1,  0,  0,  1,  1,  2,  0,  1,  1,  2,  2, ...
  -1,  0,  0,  1,  1,  2,  0,  1,  1,  2,  2, ...
  -2, -1, -1,  0,  0,  1, -1,  0,  0,  1,  1, ...
  -2, -1, -1,  0,  0,  1, -1,  0,  0,  1,  1, ...
  -3, -2, -2, -1, -1,  0, -2, -1, -1,  0,  0, ...
  -1,  0,  0,  1,  1,  2,  0,  1,  1,  2,  2, ...
  -2, -1, -1,  0,  0,  1, -1,  0,  0,  1,  1, ...
  -2, -1, -1,  0,  0,  1, -1,  0,  0,  1,  1, ...
  -3, -2, -2, -1, -1,  0, -2, -1, -1,  0,  0, ...
  -3, -2, -2, -1, -1,  0, -2, -1, -1,  0,  0, ...
  ...

Prog

(Scheme, two alternative versions)
(define (A230419 n) (- (A034968 (A025581 n)) (A034968 (A002262 n))))
;; A "stand-alone" version:
(define (A230419 n) (A230419bi (A025581 n) (A002262 n)))
(define (A230419bi 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 (- (modulo x i) (modulo y i))))))))

Crossrefs

The topmost row: A034968 (and also the leftmost column negated).

Cf. A230415 (similar array which gives the number of differing digits).

Cf. A231713 (similar array which gives the sum of absolute differences).

Sequence in context: A329767 A356018 A107502 * A146165 A308831 A277327

Adjacent sequences: A230416 A230417 A230418 * A230420 A230421 A230422

Keyword

sign,base,tabl

Author

Antti Karttunen, Nov 10 2013

Status

approved