A255329 -id:A255329 - cp's OEIS frontend

A255331 a(n) = A255329(n) - A255328(n).

-1, 0, 0, -4, 1, 0, -7, 0, -3, 1, 0, 3, 0, -6, 0, -6, 0, -3, 1, 0, 3, 0, -12, 0, 0, -5, 0, 4, 0, -6, 0, -6, 0, -3, 1, 0, 3, 0, -12, 0, 0, 7, 1, -12, 2, 0, 0, -5, 0, 4, 0, -12, 0, 0, -5, 0, 4, 0, -6, 0, -6, 0, -3, 1, 0, 3, 0, -12, 0, 0, 7, 1, -12, 2, 0, 0, 7, 1, -10, 15, 0, 0, 1, -11, 2, 0, 0, -5, 0, 4, 0, -12, 0, 0, 7, 1, -12, 2, 0, 0

Offset: 0

Antti Karttunen, Feb 21 2015

sign

a(n) = How many more nodes there are in the finite subtrees branching "right" (to the "larger side") than in the finite subtrees branching "left" (to the "smaller side") from the node n in the infinite trunk of number-of-runs beanstalk (A255056).
The edge-relation between nodes is given by A236840(child) = parent. Odd numbers are leaves, as there are no such k that A236840(k) were odd.
If A255058(n) = 1, then a(n) = 0, but also in some other cases.

			The only finite subtree starting from the node number 0 (which is 0) is the leaf 1, and it branches to the "left" (meaning that it is less than 2, which is the next node in the infinite trunk), thus the difference between the nodes in finite branches to the right vs. the nodes in finite branches to the left is -1 and a(0) = -1.
The only finite subtrees starting from the node number 1 in the infinite trunk (which is 2), are the leaves 3 and 5, of which the other one is on the "left" side and the other one on the "right" side (i.e. less than 4 and more than 4, which is the next node in the infinite trunk), thus a(1) = 1-1 = 0.
The node 11 in the infinite trunk is A255056(11) = 30. Apart from 32, which is the next node (node 12) in the infinite trunk, it has one leaf-child 31 at the "left side" (less than 32), and one leaf-child 33 (more than 32) at the "right side", and also at that side a subtree of three nodes 34 <- 38 <- 43, thus a(11) = (3+1) - 1 = 3.

Antti Karttunen, Table of n, a(n) for n = 0..8590

Partial sums: A255332.

Cf. A236840, A255058, A255328, A255329, A255330.

(define (A255331 n) (- (A255329 n) (A255328 n)))

a(n) = A255329(n) - A255328(n).

A255057 The trunk of number-of-runs beanstalk, halved: a(n) = A255056(n)/2.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 7, 9, 11, 13, 14, 15, 16, 18, 21, 23, 25, 27, 29, 30, 31, 32, 34, 37, 39, 42, 45, 47, 48, 50, 53, 55, 57, 59, 61, 62, 63, 64, 66, 69, 71, 74, 76, 78, 81, 84, 87, 90, 93, 95, 96, 98, 101, 103, 106, 109, 111, 112, 114, 117, 119, 121, 123, 125, 126, 127

Offset: 0

Antti Karttunen, Feb 14 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16142

First differences: A255337.
Characteristic function: A255339.
Cf. A255056, A255058, A255067, A255122, A255125, A255126, A255328, A255329, A255330, A255338.

a(n) = A255056(n)/2.

a(n) = A255067(A255122(n)).

A255327 a(n) = 0 if n is in the infinite trunk of "number-of-runs beanstalk" (one of the terms of A255056), otherwise number of nodes (including leaves and the node n itself) in that finite subtree of the beanstalk.

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 0, 1, 5, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 0, 1, 0, 1, 0, 1, 3, 1, 0, 1, 2, 1, 4, 1, 0, 1, 2, 1, 0, 1, 5, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 0, 1, 0, 1, 0, 1, 3, 1, 0, 1, 2, 1, 10, 1, 0, 1, 8, 1, 0, 1, 3, 1, 2, 1, 0, 1, 2, 1, 4, 1, 0, 1, 2, 1, 0, 1, 0, 1, 3, 1, 0, 1, 2, 1, 4, 1, 0, 1, 2, 1, 0, 1, 5, 1, 0, 1, 2, 1, 0

Offset: 0

Antti Karttunen, Feb 21 2015

nonn

The edge-relation between nodes is given by A236840(child) = parent. a(n) = 1 + the size of transitive closure of all children emanating from the parent at n. For any n in A255056 this would be infinite, thus such n are marked with zeros.

Odd numbers are leaves, as there are no such k that A236840(k) were odd, thus a(2n+1) = 1.

Antti Karttunen, Table of n, a(n) for n = 0..8590

Cf. A091067, A236840, A255056, A255068, A255328, A255329, A255330, A255339.

Cf. also A213727, A227643.

a(2n+1) = 1, and for even numbers 2n, if A255339(n) = 1, then a(2n) = 0, otherwise, a(2n) = 1 + sum_{k = A091067(n) .. A255068(n)} a(k).

A255330 a(n) = total number of nodes in the finite subtrees branching from the node n in the infinite trunk of "number-of-runs beanstalk" (A255056).

Original entry on oeis.org

1, 2, 0, 4, 1, 0, 7, 0, 3, 1, 0, 5, 2, 6, 0, 6, 0, 3, 1, 0, 5, 2, 12, 0, 2, 5, 0, 4, 2, 6, 0, 6, 0, 3, 1, 0, 5, 2, 12, 0, 2, 7, 1, 12, 4, 0, 2, 5, 0, 4, 2, 12, 0, 2, 5, 0, 4, 2, 6, 0, 6, 0, 3, 1, 0, 5, 2, 12, 0, 2, 7, 1, 12, 4, 0, 2, 7, 1, 10, 17, 0, 0, 1, 11, 4, 0, 2, 5, 0, 4, 2, 12, 0, 2, 7, 1, 12, 4, 0, 2, 5, 0, 4, 2, 12, 0, 2, 5, 0, 4, 2, 6, 0, 6, 0, 3, 1, 0, 5

Offset: 0

Antti Karttunen, Feb 21 2015

nonn

A255058 gives the number of branches (children) of the node n in the trunk, of which one is the next node of the infinite trunk itself. Thus, if A255058(n) = 1, then a(n) = 0.

			The edge-relation between nodes is given by A236840(child) = parent. Odd numbers are leaves, as there are no such k that A236840(k) were odd.
The node 11 in the infinite trunk is A255056(11) = 30. Apart from 32 [we have A236840(32) = 30] which is the next node (node 12) in the infinite trunk, it has a single leaf-child 31 [A236840(31) = 30] at the "left side" (less than 32), and a leaf-child 33 [A236840(33) = 30] (more than 32) at the "right side", and also at that side, a subtree of three nodes 34 <- 38 <- 43 [we have A236840(43) = 38, A236840(38) = 34 and A236840(34) = 30], thus in total there are 1+1+3 = 5 nodes in finite branches emanating from the node 11 of the infinite trunk, and a(11) = 5.

Antti Karttunen, Table of n, a(n) for n = 0..8590

Partial sums: A255333.

Cf. A091067, A236840, A255056, A255058, A255068, A255327, A255328, A255329, A255331.

(define (A255330 n) (if (zero? n) 1 (let ((k (A255057 n))) (add A255327 (A091067 k) (A255068 k)))))
;; Other code as in A255327.

a(0) = 1; a(n) = sum_{k = A091067(A255057(n)) .. A255068(A255057(n))} A255327(k).

a(n) = A255328(n) + A255329(n).

A255328 a(n) = total number of nodes in the finite subtrees branching "left" (to the "smaller side") from the node n in the infinite trunk of "number-of-runs beanstalk" (A255056).

Original entry on oeis.org

1, 1, 0, 4, 0, 0, 7, 0, 3, 0, 0, 1, 1, 6, 0, 6, 0, 3, 0, 0, 1, 1, 12, 0, 1, 5, 0, 0, 1, 6, 0, 6, 0, 3, 0, 0, 1, 1, 12, 0, 1, 0, 0, 12, 1, 0, 1, 5, 0, 0, 1, 12, 0, 1, 5, 0, 0, 1, 6, 0, 6, 0, 3, 0, 0, 1, 1, 12, 0, 1, 0, 0, 12, 1, 0, 1, 0, 0, 10, 1, 0, 0, 0, 11, 1, 0, 1, 5, 0, 0, 1, 12, 0, 1, 0, 0, 12, 1, 0, 1, 5, 0, 0, 1, 12, 0, 1, 5, 0, 0, 1, 6, 0, 6, 0, 3, 0, 0, 1

Offset: 0

Antti Karttunen, Feb 21 2015

nonn

			See example in A255330. Here we count only the nodes at the left side, thus a(11) = 1.

Antti Karttunen, Table of n, a(n) for n = 0..8590

Cf. A255056, A255057, A255327, A255329, A255330, A255331.

(define (A255328 n) (if (zero? n) 1 (add A255327 (A091067 (A255057 n)) (A255056 (+ n 1)))))
;; Other code as in A255327.

a(0) = 1; a(n) = sum_{k = A091067(A255057(n)) .. A255056(n+1)} A255327(k).

a(n) = A255330(n) - A255329(n).

A255332 Partial sums of A255331.

Original entry on oeis.org

-1, -1, -1, -5, -4, -4, -11, -11, -14, -13, -13, -10, -10, -16, -16, -22, -22, -25, -24, -24, -21, -21, -33, -33, -33, -38, -38, -34, -34, -40, -40, -46, -46, -49, -48, -48, -45, -45, -57, -57, -57, -50, -49, -61, -59, -59, -59, -64, -64, -60, -60, -72, -72, -72, -77, -77, -73, -73, -79, -79, -85, -85, -88, -87, -87, -84

Offset: 0

Antti Karttunen, Feb 21 2015

a(805) = 54 is the first positive term.
Is a(836) = -32 the last negative term?
The conspicuous "noncontinuity" which occurs in the scatter plot for the first time at n=5790 is caused by a sudden negative record at A255331(5790) = -708. Note that A255328(5790) = 708.

Antti Karttunen, Table of n, a(n) for n = 0..16140
A. Karttunen, Scatter plot of sequence plotted up to n=8590 (Drawn with the help of OEIS server's graph-utility).
A. Karttunen, Ratio A255332(n)/a(n) plotted with the help of OEIS-server

Cf. A255125, A255328, A255329, A255331, A255332, A255333.

Analogous sequences: A218789, A230409.

a(0) = -1; for n >= 1: a(n) = a(n-1) + A255331(n).

A262889 a(n) = total number of nodes in the finite subtrees branching "right" (to the "larger side") from node n in the infinite trunk (A259934) of the tree generated by edge-relation A049820(child) = parent.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 3, 0, 1, 13, 0, 0, 0, 1, 1, 2, 2, 0, 0, 0, 0, 0, 0, 2, 3, 0, 0, 5, 0, 4, 0, 1, 7, 0, 0, 7, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 5, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 3, 0, 22, 1, 0, 1, 2, 0, 6, 0, 0, 0, 0, 0, 0

Offset: 0

Antti Karttunen, Oct 04 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8107

Cf. A000005, A049820, A082284, A259934, A262686, A262697, A262888, A262890, A262894.

Cf. also A255329.

(define (A262889 n) (let ((t (A259934 n)) (u (A259934 (+ 1 n)))) (let loop ((s 0) (k (A262686 t))) (cond ((<= k u) s) ((= t (A049820 k)) (loop (+ s (A262697 k)) (- k 1))) (else (loop s (- k 1)))))))

a(n) = sum_{k = A259934(n+1) .. A262686(A259934(n))} [A049820(k) = A259934(n)] * A262697(k).
(Here [ ] stands for Iverson bracket, giving as its result 1 only when A049820(k) = A259934(n), and 0 otherwise).
Other identities. For all n >= 0:
A262890(n) = A262888(n) + a(n).

cp's OEIS Frontend

A255331 a(n) = A255329(n) - A255328(n).

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A255057 The trunk of number-of-runs beanstalk, halved: a(n) = A255056(n)/2.

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A255327 a(n) = 0 if n is in the infinite trunk of "number-of-runs beanstalk" (one of the terms of A255056), otherwise number of nodes (including leaves and the node n itself) in that finite subtree of the beanstalk.

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A255330 a(n) = total number of nodes in the finite subtrees branching from the node n in the infinite trunk of "number-of-runs beanstalk" (A255056).

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A255328 a(n) = total number of nodes in the finite subtrees branching "left" (to the "smaller side") from the node n in the infinite trunk of "number-of-runs beanstalk" (A255056).

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A255332 Partial sums of A255331.

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A262889 a(n) = total number of nodes in the finite subtrees branching "right" (to the "larger side") from node n in the infinite trunk (A259934) of the tree generated by edge-relation A049820(child) = parent.

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