A218613 -id:A218613 - cp's OEIS frontend

A218611 Positions in A218787 and A218788 of successive distinct values.

1, 5, 9, 16, 32, 59, 62, 115, 118, 208, 212, 213, 384, 389, 649, 654, 686, 703, 708, 716, 720, 723, 1310, 1326, 1328, 1338, 2236, 2369, 2422, 2432, 2452, 2458, 2466, 2476, 2486, 2488, 4545, 4601, 4625, 4627, 4637, 7811, 7817, 7819, 7826, 8287, 8511, 8526, 8631

Offset: 1

Antti Karttunen, Nov 11 2012

nonn

The terms give the positions to A213730 where for the first time a new, never-before-encountered sidetree ("tendril") appears from the side of the infinite beanstalk. See A179016, A218609 and also A218612.

Antti Karttunen, Table of n, a(n) for n = 1..72

Cf. A218609, A218610, A218612, A218613.

A218612 Numbers which are the roots of distinct not-previously-encountered side-trees ("tendrils") sprouting from the side of the infinite beanstalk (see A213730).

Original entry on oeis.org

2, 10, 22, 47, 105, 208, 224, 471, 486, 943, 966, 974, 1934, 1972, 3509, 3546, 3765, 3893, 3930, 3995, 4027, 4049, 7912, 8041, 8058, 8146, 14291, 15315, 15738, 15827, 15995, 16040, 16122, 16211, 16312, 16334, 31694, 32207, 32440, 32462, 32568, 57145, 57208

Offset: 1

Antti Karttunen, Nov 11 2012

nonn

Antti Karttunen, Table of n, a(n) for n = 1..72

A superset of A218548. Cf. A218611, A218613.

(define (A218612 n) (A213730 (A218611 n)))

a(n) = A213730(A218611(n)).

A218786 The sizes of the "tendrils" (finite side-trees sprouting at A213730, A218787) of infinite beanstalk (A179016).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 1, 2, 0, 3, 0, 0, 1, 0, 0, 1, 2, 0, 3, 0, 0, 1, 3, 0, 0, 2, 0, 2, 1, 0, 0, 0, 1, 2, 0, 3, 0, 0, 1, 3, 0, 0, 2, 0, 2, 1, 0, 3, 0, 0, 2, 0, 5, 0, 0, 6, 0, 2, 0, 1, 0, 0, 1, 2, 0, 3, 0, 0, 1, 3, 0, 0, 2, 0, 2, 1, 0, 3, 0, 0, 2

Offset: 1

Antti Karttunen, Nov 11 2012

			The first four tendrils of the beanstalk sprout at 2, 5, 6 and 9, (the first four nonzero terms of A213730) which are all leaves (i.e., in A055938), thus the first four terms of this sequence are all 0's. The next term A213730(5)=10, which is not leaf, but branches to two leaf-branches (12 and 13, as with both we have: 12-A000120(12)=10 and 13-A000120(13)=10, and both 12 and 13 are found from A055938, so the tendril at 10 is a binary tree of one internal vertex (and two leaves), i.e., \/, thus a(5)=1.

Antti Karttunen, Table of n, a(n) for n = 1..8727

Equally, a(n) = A072643(A218787(n)) = A072643(A218788(n)). Cf. A218613, A218603, A218604.

(define (A218786 n) (-1+ (A213726 (A213730 n))))

a(n) = A213726(A213730(n))-1.

A218609 Distinct values of A218787 in the order of appearance.

Original entry on oeis.org

0, 1, 2, 8, 3, 60, 172, 12, 20, 49, 54, 3016, 10096, 125744802, 101035235, 25, 1358590114, 40796719636668219, 70, 19049, 1770, 18, 7, 16261, 82682568533587123, 17, 36, 307, 315899951699378231, 3871315398, 15215587727307698, 59, 9097520004151634187729920190004140

Offset: 1

Antti Karttunen, Nov 11 2012

nonn

a(n) = A014486-index for the n-th tendril of the infinite beanstalk (A179016), which has not been encountered before, constructed with lesser numbers coming to the left, and larger to the right hand side of each branch.

Antti Karttunen, Table of n, a(n) for n = 1..72
Antti Karttunen, Illustration of how binary trees (the second rightmost column) are encoded by A014486

These are the mirror-images of binary trees given in A218610, i.e. a(n) = A057163(A218610(n)). A218613 gives the sizes of these trees. Cf. A218611.

(define (A218609 n) (A218787 (A218611 n)))

a(n) = A218787(A218611(n)).

A218610 Distinct values of A218788 in the order of appearance.

Original entry on oeis.org

0, 1, 3, 4, 2, 37, 110, 18, 11, 32, 33, 4755, 16127, 73542063, 97105360, 62, 1306632183, 39288694215537689, 193, 8150, 719, 12, 5, 13505, 246941338376004599, 13, 45, 407, 944077158106260984, 4975012595, 5738426278308884, 26, 27439590092251146768825651348524279

Offset: 1

Antti Karttunen, Nov 11 2012

nonn

a(n) = A014486-index for the n-th tendril of the infinite beanstalk (A179016), which has not been encountered before, constructed with lesser numbers coming to the right, and larger to the left hand side of each branch.

A. Karttunen, Table of n, a(n) for n = 1..72
A. Karttunen, Illustration of how binary trees (the second rightmost column) are encoded by A014486

These are the mirror-images of binary trees given in A218609, i.e. a(n) = A057163(A218609(n)). A218613 gives the sizes of these trees. Cf. A218611.

(define (A218610 n) (A218788 (A218611 n)))

a(n) = A218788(A218611(n)).

A218618 Absolute value of a(n) tells the size of the n-th side-tree ("tendril", A213730(n)) in the binary beanstalk; the sign tells on which side of the infinite trunk (A179016) it is.

Original entry on oeis.org

0, 1, -1, 1, -1, 3, 1, -1, 3, 5, -1, 1, -1, 3, 5, -1, -7, 1, 1, 3, 1, -1, 3, 5, -1, -7, 1, 1, 3, -7, 1, 1, -5, 1, -5, -3, -1, 1, -1, 3, 5, -1, -7, 1, 1, 3, -7, 1, 1, -5, 1, -5, -3, -1, -7, 1, 1, -5, 1, 11, 1, 1, 13, 1, -5, -1, 3, 1, -1, 3, 5, -1, -7, 1, 1, 3

Offset: 0

Antti Karttunen, Dec 03 2012

sign

Positive and negative terms correspond to the tendrils that sprout respectively at the left and right sides of the infinite trunk, when the beanstalk is drawn with the lesser numbers branching to the left. The absolute values give the sizes of those tendrils, with all nodes included: The leaves, the internal vertices as well as the root itself: A213730(n).

Antti Karttunen, Table of n, a(n) for n = 0..8727

Partial sums: A218785, A218789. Cf. also A218786, A218613.

(define (A218618 n) (* (expt -1 (A213730 n)) (A213727 (A213730 n))))

a(n) = -1^A213730(n) * A213727(A213730(n)).

cp's OEIS Frontend

A218611 Positions in A218787 and A218788 of successive distinct values.

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A218612 Numbers which are the roots of distinct not-previously-encountered side-trees ("tendrils") sprouting from the side of the infinite beanstalk (see A213730).

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A218786 The sizes of the "tendrils" (finite side-trees sprouting at A213730, A218787) of infinite beanstalk (A179016).

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A218609 Distinct values of A218787 in the order of appearance.

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A218610 Distinct values of A218788 in the order of appearance.

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A218618 Absolute value of a(n) tells the size of the n-th side-tree ("tendril", A213730(n)) in the binary beanstalk; the sign tells on which side of the infinite trunk (A179016) it is.

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