A218778 - OEIS

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A218778

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 right side" construction.

2, 10, 50, 210, 914, 3666, 14738, 59026, 236690, 946834, 3787922, 15151762, 60607634, 242437266, 969821330, 3879357586, 15518026898, 62072179858, 248289315986, 993157336210, 3972629941394, 15890526653586, 63562180611218, 254248729332882, 1016994991328402

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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 A218776. For more compact versions, see A218780 & A218782.

LINKS

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

A. 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 right hand side:

..........

...\1/.... Coded by A014486(A218779(1)) = A014486(1) = 2 (binary 10).

..........

.....\2/..

...\1/.... Coded by A014486(A218779(2)) = A014486(2) = 10 (bin. 1010).

..........

.\3/ \2/..

...\1/.... Coded by A014486(A218779(3)) = A014486(6) = 50 (110010).

..........

..\4/.....

.\3/.\2/..

...\1/.... Coded by A014486(A218779(4)) = A014486(16) = 210 (11010010).

..........

Thus the first four terms of this sequence are 2, 10, 50 and 210.

PROG

(Scheme with memoization macro definec from Antti Karttunen's Intseq-library):

(definec (A218778 n) (parenthesization->A014486 (tree_for_A218778 n)))

(definec (tree_for_A218778 n) (cond ((zero? n) (list)) ((= 1 n) (list (list))) (else (let ((new-tree (copy-tree (tree_for_A218778 (-1+ n))))) (add-bud-for-the-n-th-unbranching-tree-with-car-cdr-code! new-tree (A218614 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(A218779(n)). Cf. A014486, A218614, A218776, A218782, A218780. A218788.

Sequence in context: A143147 A317111 A337348 * A320521 A371605 A180266

Adjacent sequences: A218775 A218776 A218777 * A218779 A218780 A218781

KEYWORD

nonn

AUTHOR

Antti Karttunen, Nov 17 2012

STATUS

approved