cp's OEIS Frontend

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.

%I A285721 #25 Mar 28 2021 19:34:46
%S A285721 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 A285721 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 A285721 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 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.
%H A285721 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 A285721 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 A285721 Or alternatively, when n <> k, A(n,k) = 1 + A(abs(n-k),min(n,k)).
%F A285721 A(n,k) = A072030(n,k)-1.
%F A285721 As an one-dimensional sequence:
%F A285721 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 A285721 The top left 18 X 18 corner of the array:
%e A285721    0, 1, 2, 3, 4, 5, 6, 7, 8,  9, 10, 11, 12, 13, 14, 15, 16, 17
%e A285721    1, 0, 2, 1, 3, 2, 4, 3, 5,  4,  6,  5,  7,  6,  8,  7,  9,  8
%e A285721    2, 2, 0, 3, 3, 1, 4, 4, 2,  5,  5,  3,  6,  6,  4,  7,  7,  5
%e A285721    3, 1, 3, 0, 4, 2, 4, 1, 5,  3,  5,  2,  6,  4,  6,  3,  7,  5
%e A285721    4, 3, 3, 4, 0, 5, 4, 4, 5,  1,  6,  5,  5,  6,  2,  7,  6,  6
%e A285721    5, 2, 1, 2, 5, 0, 6, 3, 2,  3,  6,  1,  7,  4,  3,  4,  7,  2
%e A285721    6, 4, 4, 4, 4, 6, 0, 7, 5,  5,  5,  5,  7,  1,  8,  6,  6,  6
%e A285721    7, 3, 4, 1, 4, 3, 7, 0, 8,  4,  5,  2,  5,  4,  8,  1,  9,  5
%e A285721    8, 5, 2, 5, 5, 2, 5, 8, 0,  9,  6,  3,  6,  6,  3,  6,  9,  1
%e A285721    9, 4, 5, 3, 1, 3, 5, 4, 9,  0, 10,  5,  6,  4,  2,  4,  6,  5
%e A285721   10, 6, 5, 5, 6, 6, 5, 5, 6, 10,  0, 11,  7,  6,  6,  7,  7,  6
%e A285721   11, 5, 3, 2, 5, 1, 5, 2, 3,  5, 11,  0, 12,  6,  4,  3,  6,  2
%e A285721   12, 7, 6, 6, 5, 7, 7, 5, 6,  6,  7, 12,  0, 13,  8,  7,  7,  6
%e A285721   13, 6, 6, 4, 6, 4, 1, 4, 6,  4,  6,  6, 13,  0, 14,  7,  7,  5
%e A285721   14, 8, 4, 6, 2, 3, 8, 8, 3,  2,  6,  4,  8, 14,  0, 15,  9,  5
%e A285721   15, 7, 7, 3, 7, 4, 6, 1, 6,  4,  7,  3,  7,  7, 15,  0, 16,  8
%e A285721   16, 9, 7, 7, 6, 7, 6, 9, 9,  6,  7,  6,  7,  7,  9, 16,  0, 17
%e A285721   17, 8, 5, 5, 6, 2, 6, 5, 1,  5,  6,  2,  6,  5,  5,  8, 17,  0
%o A285721 (Scheme)
%o A285721 (define (A285721 n) (A285721bi (A002260 n) (A004736 n)))
%o A285721 (define (A285721bi row col) (cond ((= row col) 0) ((> row col) (+ 1 (A285721bi (- row col) col))) (else (+ 1 (A285721bi row (- col row))))))
%o A285721 ;; Alternatively:
%o A285721 (define (A285721bi row col) (if (= row col) 0 (+ 1 (A285721bi (abs (- row col)) (min col row)))))
%o A285721 ;; Another implementation, as an one-dimensional sequence:
%o A285721 (definec (A285721 n) (if (zero? (A285722 n)) 0 (+ 1 (A285721 (A285722 n)))))
%o A285721 (Python)
%o A285721 def A(n, k): return 0 if n==k else 1 + A(abs(n - k), min(n, k))
%o A285721 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 A285721 One less than A072030.
%Y A285721 Row 2 & column 2: A028242 (but with starting offset 1).
%Y A285721 Row 3 & column 3 (from zero onward) seems to be A226576.
%Y A285721 Cf. A003989, A285722, A285732.
%Y A285721 Compare also to arrays A049834, A113881, A219158.
%K A285721 nonn,tabl
%O A285721 1,4
%A A285721 _Antti Karttunen_, May 03 2017