A285721 - OEIS

%I #25 Mar 28 2021 19:34:46

%S 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,

%T 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,

%U 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

%N 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.

%H Antti Karttunen, <a href="/A285721/b285721.txt">Table of n, a(n) for n = 1..10585; the first 145 antidiagonals of the array</a>

%F 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).

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

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

%F As an one-dimensional sequence:

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

%e The top left 18 X 18 corner of the array:

%e    0, 1, 2, 3, 4, 5, 6, 7, 8,  9, 10, 11, 12, 13, 14, 15, 16, 17

%e    1, 0, 2, 1, 3, 2, 4, 3, 5,  4,  6,  5,  7,  6,  8,  7,  9,  8

%e    2, 2, 0, 3, 3, 1, 4, 4, 2,  5,  5,  3,  6,  6,  4,  7,  7,  5

%e    3, 1, 3, 0, 4, 2, 4, 1, 5,  3,  5,  2,  6,  4,  6,  3,  7,  5

%e    4, 3, 3, 4, 0, 5, 4, 4, 5,  1,  6,  5,  5,  6,  2,  7,  6,  6

%e    5, 2, 1, 2, 5, 0, 6, 3, 2,  3,  6,  1,  7,  4,  3,  4,  7,  2

%e    6, 4, 4, 4, 4, 6, 0, 7, 5,  5,  5,  5,  7,  1,  8,  6,  6,  6

%e    7, 3, 4, 1, 4, 3, 7, 0, 8,  4,  5,  2,  5,  4,  8,  1,  9,  5

%e    8, 5, 2, 5, 5, 2, 5, 8, 0,  9,  6,  3,  6,  6,  3,  6,  9,  1

%e    9, 4, 5, 3, 1, 3, 5, 4, 9,  0, 10,  5,  6,  4,  2,  4,  6,  5

%e   10, 6, 5, 5, 6, 6, 5, 5, 6, 10,  0, 11,  7,  6,  6,  7,  7,  6

%e   11, 5, 3, 2, 5, 1, 5, 2, 3,  5, 11,  0, 12,  6,  4,  3,  6,  2

%e   12, 7, 6, 6, 5, 7, 7, 5, 6,  6,  7, 12,  0, 13,  8,  7,  7,  6

%e   13, 6, 6, 4, 6, 4, 1, 4, 6,  4,  6,  6, 13,  0, 14,  7,  7,  5

%e   14, 8, 4, 6, 2, 3, 8, 8, 3,  2,  6,  4,  8, 14,  0, 15,  9,  5

%e   15, 7, 7, 3, 7, 4, 6, 1, 6,  4,  7,  3,  7,  7, 15,  0, 16,  8

%e   16, 9, 7, 7, 6, 7, 6, 9, 9,  6,  7,  6,  7,  7,  9, 16,  0, 17

%e   17, 8, 5, 5, 6, 2, 6, 5, 1,  5,  6,  2,  6,  5,  5,  8, 17,  0

%o (Scheme)

%o (define (A285721 n) (A285721bi (A002260 n) (A004736 n)))

%o (define (A285721bi row col) (cond ((= row col) 0) ((> row col) (+ 1 (A285721bi (- row col) col))) (else (+ 1 (A285721bi row (- col row))))))

%o ;; Alternatively:

%o (define (A285721bi row col) (if (= row col) 0 (+ 1 (A285721bi (abs (- row col)) (min col row)))))

%o ;; Another implementation, as an one-dimensional sequence:

%o (definec (A285721 n) (if (zero? (A285722 n)) 0 (+ 1 (A285721 (A285722 n)))))

%o (Python)

%o def A(n, k): return 0 if n==k else 1 + A(abs(n - k), min(n, k))

%o for n in range(1, 21): print([A(n - k + 1, k) for k in range(1, n + 1)]) # _Indranil Ghosh_, May 03 2017

%Y One less than A072030.

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

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

%Y Cf. A003989, A285722, A285732.

%Y Compare also to arrays A049834, A113881, A219158.

%K nonn,tabl

%O 1,4

%A _Antti Karttunen_, May 03 2017