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

Author

Antti Karttunen, Nov 10 2013

Keywords

• nonn
• base
• tabl

Examples

			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.
		

Links

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

Crossrefs

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

Cf. A231714, A060130, A055881.

Programs

Scheme

(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)))))))

Formula

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).