A265751 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).

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, 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, 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

Offset: 0

Author

Antti Karttunen, Dec 21 2015

Keywords

• nonn
• tabl

Comments

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), ...

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.

Examples

			The top left corner of the array:
   0,  1,  3,  5,  7,  0,  0,  0,  0
   1,  3,  5,  7,  0,  0,  0,  0,  0
   2,  6,  9, 11, 13,  0,  0,  0,  0
   3,  5,  7,  0,  0,  0,  0,  0,  0
   4,  8,  0,  0,  0,  0,  0,  0,  0
   5,  7,  0,  0,  0,  0,  0,  0,  0
   6,  9, 11, 13,  0,  0,  0,  0,  0
   7,  0,  0,  0,  0,  0,  0,  0,  0
   8,  0,  0,  0,  0,  0,  0,  0,  0
   9, 11, 13,  0,  0,  0,  0,  0,  0
  10, 14, 20,  0,  0,  0,  0,  0,  0
  11, 13,  0,  0,  0,  0,  0,  0,  0
  12, 18, 22, 25,  0,  0,  0,  0,  0
  13,  0,  0,  0,  0,  0,  0,  0,  0
  14, 20,  0,  0,  0,  0,  0,  0,  0
  15, 17, 19,  0,  0,  0,  0,  0,  0
  16, 24,  0,  0,  0,  0,  0,  0,  0
  17, 19,  0,  0,  0,  0,  0,  0,  0
  18, 22, 25,  0,  0,  0,  0,  0,  0
  19,  0,  0,  0,  0,  0,  0,  0,  0
  20,  0,  0,  0,  0,  0,  0,  0,  0
  21, 23, 27, 29, 31, 35, 37,  0,  0
  22, 25,  0,  0,  0,  0,  0,  0,  0
  23, 27, 29, 31, 35, 37,  0,  0,  0
  ...
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.
		

Crossrefs

Cf. also A000005, A045765, A060990.

Column 0: A001477, Column 1: A082284.

Cf. A266111 (number of significant terms on each row, without the trailing zeros).

Cf. A266116 (the rightmost term before trailing zeros).

See also array A263271 constructed in the same way, but obtained by following always the largest child A262686, instead of the smallest child A082284.

Cf. also tree A263267 (and its illustration).

Programs

Scheme

(define (A265751 n) (A265751bi (A002262 n) (A025581 n)))
(define (A265751bi row col) (cond ((zero? col) row) ((A082284 row) => (lambda (lad) (if (zero? lad) lad (A265751bi lad (- col 1)))))))
;; Alternatively:
(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))))))

Formula

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).

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)).