A285721 Square array read by antidiagonals: A(n,k) = number of steps in simple Euclidean algorithm for gcd(n,k) to reach the termination test n=k, read by antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

0, 1, 1, 2, 0, 2, 3, 2, 2, 3, 4, 1, 0, 1, 4, 5, 3, 3, 3, 3, 5, 6, 2, 3, 0, 3, 2, 6, 7, 4, 1, 4, 4, 1, 4, 7, 8, 3, 4, 2, 0, 2, 4, 3, 8, 9, 5, 4, 4, 5, 5, 4, 4, 5, 9, 10, 4, 2, 1, 4, 0, 4, 1, 2, 4, 10, 11, 6, 5, 5, 4, 6, 6, 4, 5, 5, 6, 11, 12, 5, 5, 3, 5, 3, 0, 3, 5, 3, 5, 5, 12, 13, 7, 3, 5, 1, 2, 7, 7, 2, 1, 5, 3, 7, 13, 14, 6, 6, 2, 6, 3, 5, 0, 5, 3, 6, 2, 6, 6, 14

Offset: 1

Author

Antti Karttunen, May 03 2017

Keywords

• nonn
• tabl

Examples

			The top left 18 X 18 corner of the array:
   0, 1, 2, 3, 4, 5, 6, 7, 8,  9, 10, 11, 12, 13, 14, 15, 16, 17
   1, 0, 2, 1, 3, 2, 4, 3, 5,  4,  6,  5,  7,  6,  8,  7,  9,  8
   2, 2, 0, 3, 3, 1, 4, 4, 2,  5,  5,  3,  6,  6,  4,  7,  7,  5
   3, 1, 3, 0, 4, 2, 4, 1, 5,  3,  5,  2,  6,  4,  6,  3,  7,  5
   4, 3, 3, 4, 0, 5, 4, 4, 5,  1,  6,  5,  5,  6,  2,  7,  6,  6
   5, 2, 1, 2, 5, 0, 6, 3, 2,  3,  6,  1,  7,  4,  3,  4,  7,  2
   6, 4, 4, 4, 4, 6, 0, 7, 5,  5,  5,  5,  7,  1,  8,  6,  6,  6
   7, 3, 4, 1, 4, 3, 7, 0, 8,  4,  5,  2,  5,  4,  8,  1,  9,  5
   8, 5, 2, 5, 5, 2, 5, 8, 0,  9,  6,  3,  6,  6,  3,  6,  9,  1
   9, 4, 5, 3, 1, 3, 5, 4, 9,  0, 10,  5,  6,  4,  2,  4,  6,  5
  10, 6, 5, 5, 6, 6, 5, 5, 6, 10,  0, 11,  7,  6,  6,  7,  7,  6
  11, 5, 3, 2, 5, 1, 5, 2, 3,  5, 11,  0, 12,  6,  4,  3,  6,  2
  12, 7, 6, 6, 5, 7, 7, 5, 6,  6,  7, 12,  0, 13,  8,  7,  7,  6
  13, 6, 6, 4, 6, 4, 1, 4, 6,  4,  6,  6, 13,  0, 14,  7,  7,  5
  14, 8, 4, 6, 2, 3, 8, 8, 3,  2,  6,  4,  8, 14,  0, 15,  9,  5
  15, 7, 7, 3, 7, 4, 6, 1, 6,  4,  7,  3,  7,  7, 15,  0, 16,  8
  16, 9, 7, 7, 6, 7, 6, 9, 9,  6,  7,  6,  7,  7,  9, 16,  0, 17
  17, 8, 5, 5, 6, 2, 6, 5, 1,  5,  6,  2,  6,  5,  5,  8, 17,  0
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10585; the first 145 antidiagonals of the array

Crossrefs

One less than A072030.

Row 2 & column 2: A028242 (but with starting offset 1).

Row 3 & column 3 (from zero onward) seems to be A226576.

Cf. A003989, A285722, A285732.

Compare also to arrays A049834, A113881, A219158.

Programs

Python

def A(n, k): return 0 if n==k else 1 + A(abs(n - k), min(n, k))
for n in range(1, 21): print([A(n - k + 1, k) for k in range(1, n + 1)]) # Indranil Ghosh, May 03 2017

Scheme

(define (A285721 n) (A285721bi (A002260 n) (A004736 n)))
(define (A285721bi row col) (cond ((= row col) 0) ((> row col) (+ 1 (A285721bi (- row col) col))) (else (+ 1 (A285721bi row (- col row))))))
;; Alternatively:
(define (A285721bi row col) (if (= row col) 0 (+ 1 (A285721bi (abs (- row col)) (min col row)))))
;; Another implementation, as an one-dimensional sequence:
(definec (A285721 n) (if (zero? (A285722 n)) 0 (+ 1 (A285721 (A285722 n)))))

Formula

If n = k, then A(n,k) = 0, if n > k, then A(n,k) = 1 + A(n-k,k), otherwise [when n < k], A(n,k) = 1 + A(n,k-n).

Or alternatively, when n <> k, A(n,k) = 1 + A(abs(n-k),min(n,k)).

A(n,k) = A072030(n,k)-1.

As an one-dimensional sequence:

a(n) = 0 if A285722(n) = 0, otherwise a(n) = 1 + a(A285722(n)). [Here A285722 is also used as an one-dimensional sequence.]