A173601 -id:A173601 - cp's OEIS frontend

A218604 a(n) = A173601(n) - A179016(n).

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

Offset: 0

Antti Karttunen, Nov 03 2012

nonn

For all n, the following holds: A213708(n) <= A179016(n) <= A173601(n). This sequence gives the distance of the node n in the infinite trunk of beanstalk (A179016(n)) from the greater edge of the A086876(n) wide window which it at that point must pass through.

The increasing steps seem to be quite constrained in their magnitude, compared to the decreasing steps. (This depends on how the "tendrils",i.e. the finite side-trees on the other side of the infinite trunk grow and reach their tops).

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

Positions of zeros: A218608, A218606.

Cf. A086876, A173601, A179016, A218603.

(define (A218604 n) (- (A173601 n) (A179016 n)))

a(n) = A086876(n) - A218603(n) - 1.

Offset changed because of the changed offset of A179016 - Antti Karttunen, Nov 10 2012

A218608 The positions of zeros in A218604, i.e., those integers i for which A179016(i) = A173601(i).

Original entry on oeis.org

0, 1, 2, 4, 6, 7, 9, 10, 12, 14, 15, 21, 23, 24, 38, 40, 41, 61, 62, 63, 64, 68, 70, 71, 91, 92, 93, 94, 104, 105, 106, 107, 113, 122, 124, 125, 145, 146, 147, 148, 158, 159, 160, 161, 167, 182, 183, 184, 185, 191, 202, 220, 222, 223, 243, 244, 245, 246, 256

Offset: 0

Antti Karttunen, Nov 10 2012

nonn

These are the points i for which the corresponding node in the infinite trunk of beanstalk (A179016(i)) is at the greatest possible position of the allotted "window" which it at that point must pass through, i.e. there are no leaves or side-trees at its right (greater) side at these points. (See comments at A218604.)

Antti Karttunen, Table of n, a(n) for n = 0..7730

Apart from zero, a subset of A213733. Cf. A218607, A218606.

A179016 The infinite trunk of binary beanstalk: The only infinite sequence such that a(n-1) = a(n) - number of 1's in binary representation of a(n).

Original entry on oeis.org

0, 1, 3, 4, 7, 8, 11, 15, 16, 19, 23, 26, 31, 32, 35, 39, 42, 46, 49, 53, 57, 63, 64, 67, 71, 74, 78, 81, 85, 89, 94, 97, 101, 104, 109, 112, 116, 120, 127, 128, 131, 135, 138, 142, 145, 149, 153, 158, 161, 165, 168, 173, 176, 180, 184, 190, 193, 197, 200, 205, 209

Offset: 0

Carl R. White, Jun 24 2010

a(n) tells in what number we end in n steps, when we start climbing up the infinite trunk of the "binary beanstalk" from its root (zero). The name "beanstalk" is due to Antti Karttunen.

There are many finite sequences such as 0,1,2; 0,1,3,4,7,9; etc. obeying the same condition (see A218254) and as the length increases, so (necessarily) does the similarity to this infinite sequence.

Alois P. Heinz and Antti Karttunen, Table of n, a(n) for n = 0..16405 (first 1000 terms from Alois P. Heinz)
Paul Tek, Illustration of the first terms

Cf. A000120, A010062, A011371, A213710, A213711, A213717, A213730, A213731, A218600, A218616, A218789, A233271, A218602, A054429. First differences: A213712, complement: A213713.
A subsequence of A005187, i.e., a(n) = A005187(A213715(n)). For all n,
A071542(a(n)) = n, and furthermore A213708(n) <= a(n) <= A173601(n). (Cf. A218603, A218604).
Rows of A218254, when reversed, converge towards this sequence.
Cf. A276623, A219648, A219666, A255056, A276573, A276583, A276613 for analogous constructions, and also A259934.

TakeWhile[Reverse@ NestWhileList[# - DigitCount[#, 2, 1] &, 10^3, # > 0 &], # <= 209 &] (* Michael De Vlieger, Sep 12 2016 *)

a(0)=0, a(1)=1, and for n > 1, if n = A218600(A213711(n)) then a(n) = (2^A213711(n)) - 1, and in other cases, a(n) = a(n+1) - A213712(n+1). (This formula is based on Carl White's observation that this iterated/converging path must pass through each (2^n)-1. However, it would be very interesting to know whether the sequence admits more traditional recurrence(s), referring to previous, not to further terms in the sequence in their definition!) - Antti Karttunen, Oct 26 2012
a(n) = A218616(A218602(n)). - Antti Karttunen, Mar 04 2013
a(n) = A054429(A233271(A218602(n))). - Antti Karttunen, Dec 12 2013

Starting offset changed from 1 to 0 by Antti Karttunen, Nov 05 2012

A213708 a(n) is the least inverse of A071542, i.e., minimal i such that A071542(i) = n.

Original entry on oeis.org

0, 1, 2, 4, 6, 8, 10, 12, 16, 18, 20, 24, 28, 32, 34, 36, 40, 44, 48, 52, 56, 60, 64, 66, 68, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 126, 128, 130, 132, 136, 140, 144, 148, 152, 156, 160, 164, 168, 172, 176, 180, 184, 190, 192, 196, 200, 204, 208, 212, 216, 222, 226, 232, 238, 244, 250, 256, 258, 260, 264, 268, 272, 276

Offset: 0

Antti Karttunen, Oct 24 2012

nonn

Also the positions in A071542 where new records appear, record values appearing in the ascending order, i.e., as A001477 (because A071542 is a monotone and surjective function).

Antti Karttunen, Table of n, a(n) for n = 0..16500

Cf. A173601 for the greatest inverse. A086876 gives the first differences.

Cf. A179016, A213730.

Function[s, Map[FirstPosition[s, #] &, Union@ s]]@ Table[-1 + Length@ NestWhileList[# - DigitCount[#, 2, 1] &, n, # > 0 &], {n, 276}] // Flatten (* Michael De Vlieger, Jul 16 2017 *)

A219655 Greatest inverse of A219652; a(n) = maximal i such that A219652(i) = n.

Original entry on oeis.org

0, 1, 3, 5, 7, 11, 15, 19, 23, 25, 29, 33, 37, 41, 47, 51, 55, 59, 65, 71, 77, 83, 89, 95, 101, 107, 115, 119, 121, 125, 129, 133, 137, 143, 147, 151, 155, 161, 167, 173, 179, 185, 191, 197, 203, 211, 217, 225, 233, 239, 243, 247, 251, 257, 263, 269, 275, 281

Offset: 0

Antti Karttunen, Nov 25 2012

nonn

A. Karttunen, Table of n, a(n) for n = 0..10080

Cf. A219653 for the least inverse. A219654 gives the first differences.

This sequence is based on factorial number system: A007623. Analogous sequence for binary system: A173601, for Zeckendorf expansion: A219645.

a(n) = A219653(n) + A219654(n) - 1.

A086876 Run lengths in A071542.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 2, 4, 2, 2, 4, 4, 4, 2, 2, 4, 4, 4, 4, 4, 4, 4, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 6, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 6, 2, 4, 4, 4, 4, 4, 4, 6, 4, 6, 6, 6, 6, 6, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 6, 2, 4, 4, 4, 4, 4, 4, 6, 4, 6, 6, 6

Offset: 0

Ralf Stephan, Aug 21 2003

All a(n) are even for n>1.
Although this can be viewed as a list, the indexing still starts from zero, because a(n) tells from how many starting values one can end to 0 in n steps, with the iterative process described in A071542 (if going around in 0->0 loop is disallowed). I.e., a(n) gives the number of all nodes (whether internal or leaves) in "beanstalk" (see A179016) from which the distance to the root (zero) is n.
Records occur at positions { 1,2,7,37,122,... } which correspond to run start positions { 2,4,16,126,512,... } in A071542.

			There is only one way to reach 0 in 0 steps from anywhere, and that is from 0 itself.
There is only one way to reach 0 in 1 steps from anywhere (with no 0->0 transition allowed), and that is from 1, as 1-A000120(1)=0.
There are two ways to reach 0 in 2 steps, from 2, as 2-A000120(2)=1, and 1-A000120(1)=0, and from 3, as 3-A000120(3)=1, and 1-A000120(1)=0.
Thus a(0)=a(1)=1 and a(2)=2.

Antti Karttunen, Table of n, a(n) for n = 0..8728

Essentially the first differences of both A173601 and A213708.

e1(n)=sum(k=0, floor(log(n)/log(2)), bittest(n, k))
f(n)=local(c); c=0; while(n, n=n-e1(n); c=c+1); c
p=1; r=1; for(n=1, 150, c=0; while(f(r) == p, r=r+1; c=c+1); p=f(r); print1(c", "))

Changed the starting offset by prepending a(0)=1 (with the indexing of the rest of terms thus not changed), as A071542 now contains an initial zero. - Antti Karttunen, Nov 02 2012

A218603 a(n) = A179016(n) - A213708(n).

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 3, 0, 1, 3, 2, 3, 0, 1, 3, 2, 2, 1, 1, 1, 3, 0, 1, 3, 2, 2, 1, 1, 1, 2, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 3, 2, 2, 1, 1, 1, 2, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 3, 5, 3, 5, 4, 2, 3, 5, 0, 1, 3, 2, 2, 1, 1, 1, 2, 1, 1, 0, 1, 0, 0, 0, 0, 1

Offset: 0

Antti Karttunen, Nov 03 2012

nonn

For all n, the following holds: A213708(n) <= A179016(n) <= A173601(n). This sequence gives the distance of the node n in the infinite trunk of beanstalk (A179016(n)) from the lesser edge of the A086876(n) wide window which it at that point must pass through.

The increasing steps seem to be quite constrained in their magnitude, compared to the decreasing steps. (This depends on how the "tendrils", i.e. the finite side-trees on the other side of the infinite trunk grow and reach their tops).

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

Positions of zeros: A218607, A218605.

Cf. A086876, A179016, A213708, A218604.

(define (A218603 n) (- (A179016 n) (A213708 n)))

a(n) = A086876(n) - A218604(n) - 1.

Offset changed because of the changed offset of A179016 - Antti Karttunen, Nov 10 2012

A219645 Greatest inverse of A219642; a(n) = maximal i such that A219642(i) = n.

Original entry on oeis.org

0, 1, 2, 4, 6, 7, 9, 12, 14, 17, 20, 22, 25, 28, 31, 33, 35, 38, 41, 44, 46, 49, 53, 54, 56, 59, 62, 65, 67, 70, 74, 77, 80, 83, 88, 90, 93, 96, 99, 101, 104, 108, 111, 114, 117, 122, 125, 129, 133, 137, 142, 143, 145, 148, 151, 154, 156, 159, 163, 166, 169

Offset: 0

Antti Karttunen, Nov 24 2012

nonn

A. Karttunen, Table of n, a(n) for n = 0..10000

Cf. A219643 for the least inverse. A219644 gives the first differences.

This sequence is based on Fibonacci number system (Zeckendorf expansion): A014417. Analogous sequence for binary system: A173601, for factorial number system: A219655.

a(n) = A219643(n)+A219644(n)-1.

A255055 Greatest inverse of A255072; a(n) = largest k such that A255072(k) = n.

Original entry on oeis.org

0, 2, 5, 6, 10, 13, 14, 18, 22, 26, 29, 30, 34, 38, 43, 46, 50, 54, 58, 61, 62, 66, 70, 75, 78, 85, 90, 94, 98, 102, 107, 110, 114, 118, 122, 125, 126, 130, 134, 139, 142, 149, 154, 158, 165, 171, 175, 181, 186, 190, 194, 198, 203, 206, 213, 218, 222, 226, 230, 235, 238, 242, 246, 250, 253, 254, 258

Offset: 0

Antti Karttunen, Feb 14 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16143

Cf. A255053, A255054, A255072.

Analogous sequences: A173601, A219645, A219655.

a(0) = 0; for n > 0, a(n) = A255054(n) + a(n-1).
Other identities. For all n >= 0:
a(n) = A255053(n) + A255054(n) - 1.
a(n) = A255056(n) + A255124(n).

cp's OEIS Frontend

A218604 a(n) = A173601(n) - A179016(n).

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A218608 The positions of zeros in A218604, i.e., those integers i for which A179016(i) = A173601(i).

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A179016 The infinite trunk of binary beanstalk: The only infinite sequence such that a(n-1) = a(n) - number of 1's in binary representation of a(n).

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A213708 a(n) is the least inverse of A071542, i.e., minimal i such that A071542(i) = n.

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Mathematica

A219655 Greatest inverse of A219652; a(n) = maximal i such that A219652(i) = n.

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A086876 Run lengths in A071542.

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A218603 a(n) = A179016(n) - A213708(n).

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A219645 Greatest inverse of A219642; a(n) = maximal i such that A219642(i) = n.

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A255055 Greatest inverse of A255072; a(n) = largest k such that A255072(k) = n.

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