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%I #14 Dec 23 2015 14:08:06
%S 0,1,1,3,3,2,5,5,6,3,7,7,9,5,4,0,0,11,7,8,5,0,0,13,0,0,7,6,0,0,0,0,0,
%T 0,9,7,0,0,0,0,0,0,11,0,8,0,0,0,0,0,0,13,0,0,9,0,0,0,0,0,0,0,0,0,11,
%U 10,0,0,0,0,0,0,0,0,0,13,14,11,0,0,0,0,0,0,0,0,0,0,20,13,12,0,0,0,0,0,0,0,0,0,0,0,0,18,13,0,0,0,0,0,0,0,0,0,0,0,0,22,0,14
%N Square array A(row,col): A(row,0) = row and for col >= 1, if A082284(row) is 0, then A(row,col) = 0, otherwise A(row,col) = A(A082284(row),col-1).
%C The square array A(row>=0, col>=0) is read by downwards antidiagonals as: A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), A(0,3), A(1,2), A(2,1), A(3,0), ...
%C Each row n lists all the nodes in A263267-tree that one encounters when one starts from node with number n and always chooses the smallest possible child of it [given by A082284(n)], and then the smallest possible child of that child, etc, until a leaf-child (one of the terms of A045765) is encountered, after which the rest of the row contains only zeros.
%F A(row,0) = row and for col >= 1, if A082284(row) is 0, then A(row,col) = 0, otherwise A(row,col) = A(A082284(row),col-1).
%F A(0,0) = 0, A(0,1) = 1; if col = 0, A(row,0) = row; and for col > 0, if A(row,col-1) = 0, then A(row,col) = 0, otherwise A(row,col) = A082284(A(row,col-1)).
%e The top left corner of the array:
%e 0, 1, 3, 5, 7, 0, 0, 0, 0
%e 1, 3, 5, 7, 0, 0, 0, 0, 0
%e 2, 6, 9, 11, 13, 0, 0, 0, 0
%e 3, 5, 7, 0, 0, 0, 0, 0, 0
%e 4, 8, 0, 0, 0, 0, 0, 0, 0
%e 5, 7, 0, 0, 0, 0, 0, 0, 0
%e 6, 9, 11, 13, 0, 0, 0, 0, 0
%e 7, 0, 0, 0, 0, 0, 0, 0, 0
%e 8, 0, 0, 0, 0, 0, 0, 0, 0
%e 9, 11, 13, 0, 0, 0, 0, 0, 0
%e 10, 14, 20, 0, 0, 0, 0, 0, 0
%e 11, 13, 0, 0, 0, 0, 0, 0, 0
%e 12, 18, 22, 25, 0, 0, 0, 0, 0
%e 13, 0, 0, 0, 0, 0, 0, 0, 0
%e 14, 20, 0, 0, 0, 0, 0, 0, 0
%e 15, 17, 19, 0, 0, 0, 0, 0, 0
%e 16, 24, 0, 0, 0, 0, 0, 0, 0
%e 17, 19, 0, 0, 0, 0, 0, 0, 0
%e 18, 22, 25, 0, 0, 0, 0, 0, 0
%e 19, 0, 0, 0, 0, 0, 0, 0, 0
%e 20, 0, 0, 0, 0, 0, 0, 0, 0
%e 21, 23, 27, 29, 31, 35, 37, 0, 0
%e 22, 25, 0, 0, 0, 0, 0, 0, 0
%e 23, 27, 29, 31, 35, 37, 0, 0, 0
%e ...
%e Starting from n = 21, we get the following chain: 21 -> 23 -> 27 -> 29 -> 31 -> 35 -> 37, with A082284 iterated 6 times before the final nonzero term 37 (for which A060990(37) = A082284(37) = 0) is encountered. Thus the row 21 of array contains terms 21, 23, 27, 29, 31, 35, 37, followed by an infinite number of zeros.
%o (Scheme)
%o (define (A265751 n) (A265751bi (A002262 n) (A025581 n)))
%o (define (A265751bi row col) (cond ((zero? col) row) ((A082284 row) => (lambda (lad) (if (zero? lad) lad (A265751bi lad (- col 1)))))))
%o ;; Alternatively:
%o (define (A265751bi row col) (cond ((zero? col) row) ((and (zero? row) (= 1 col)) 1) ((zero? (A265751bi row (- col 1))) 0) (else (A082284 (A265751bi row (- col 1))))))
%Y Cf. also A000005, A045765, A060990.
%Y Column 0: A001477, Column 1: A082284.
%Y Cf. A266111 (number of significant terms on each row, without the trailing zeros).
%Y Cf. A266116 (the rightmost term before trailing zeros).
%Y See also array A263271 constructed in the same way, but obtained by following always the largest child A262686, instead of the smallest child A082284.
%Y Cf. also tree A263267 (and its illustration).
%K nonn,tabl
%O 0,4
%A _Antti Karttunen_, Dec 21 2015
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