A218776 A014486-codes for the Beanstalk-tree growing one natural number at time, starting from the tree of one internal node (1), with the "lesser numbers to the left hand side" construction.

2, 12, 50, 204, 818, 3298, 13202, 52834, 211346, 845586, 3382418, 13531282, 54125714, 216503058, 866012306, 3464049426, 13856197778, 55424792722, 221699171474, 886796698770, 3547186799762, 14188747200658, 56754988803218, 227019955225746, 908079820907666

Offset

1,1

Comments

The active middle region of the triangle (see attached "Wolframesque" illustration) corresponds to the area where the growing tip(s) of the beanstalk are located. Successively larger "turbulences" occurring in that area appear approximately at the row numbers given by A218548. The larger tendrils (the finite side-trees) are, the longer there is vacillation in the direction of the growing region, which lasts until the growing tip of the infinite stem (A179016) has passed the topmost tips of the tendril. See also A218612.

These are the mirror-images (in binary tree sense) of the terms in sequence A218778. For more compact versions, see A218780 & A218782.

Links

Antti Karttunen, Table of n, a(n) for n = 1..256

Antti Karttunen, Terms a(1)-a(4096) drawn as binary strings, in Wolframesque fashion.

Example

Illustration how the growing beanstalk-tree produces the first four terms of this sequence. In this variant, the lesser numbers come to the left hand side:
..........
...\1/.... Coded by A014486(A218777(1)) = A014486(1) = 2 (binary 10).
..........
..........
.\2/......
...\1/.... Coded by A014486(A218777(2)) = A014486(3) = 12 (bin. 1100).
..........
..........
.\2/ \3/..
...\1/.... Coded by A014486(A218777(3)) = A014486(6) = 50 (110010).
..........
..........
....\4/...
.\2/.\3/..
...\1/.... Coded by A014486(A218777(4)) = A014486(15) = 204 (11001100).
..........
Thus the first four terms of this sequence are 2, 12, 50 and 204.

Prog

(Scheme with memoization macro definec from Antti Karttunen's Intseq-library):
(definec (A218776 n) (parenthesization->A014486 (tree_for_A218776 n)))
(definec (tree_for_A218776 n) (cond ((zero? n) (list)) ((= 1 n) (list (list))) (else (let ((new-tree (copy-tree (tree_for_a218776 (-1+ n))))) (add-bud-for-the-n-th-unbranching-tree-with-car-cdr-code! new-tree (A218615 n))))))
(define (add-bud-for-the-n-th-unbranching-tree-with-car-cdr-code! tree n) (let loop ((n n) (t tree)) (cond ((zero? n) (list)) ((= n 1) (list (list))) ((= n 2) (set-cdr! t (list (list)))) ((= n 3) (set-car! t (list (list)))) ((even? n) (loop (/ n 2) (cdr t))) (else (loop (/ (- n 1) 2) (car t))))) tree)
(define (copy-tree bt) (cond ((not (pair? bt)) bt) (else (cons (copy-tree (car bt)) (copy-tree (cdr bt))))))
(define (parenthesization->a014486 p) (let loop ((s 0) (p p)) (if (null? p) s (let* ((x (parenthesization->a014486 (car p))) (w (binwidth x))) (loop (+ (* s (expt 2 (+ w 2))) (expt 2 (1+ w)) (* 2 x)) (cdr p))))))
(define (binwidth n) (let loop ((n n) (i 0)) (if (zero? n) i (loop (floor->exact (/ n 2)) (1+ i))))) ;; (binwidth n) = A029837(n+1).

Crossrefs

a(n) = A014486(A218777(n)). Cf. A014486, A218615, A218780, A218782, A218787, A218778.

Sequence in context: A202789 A129743 A115243 * A241683 A341546 A012423

Adjacent sequences: A218773 A218774 A218775 * A218777 A218778 A218779

Keyword

nonn

Author

Antti Karttunen, Nov 17 2012

Status

approved