A213712 -id:A213712 - cp's OEIS frontend

A213718 n occurs A213712(n) times.

1, 2, 2, 3, 4, 4, 4, 5, 6, 6, 6, 7, 7, 7, 7, 8, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 12, 12, 12, 12, 12, 13, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 19, 19, 19, 19, 20, 20, 20, 20, 21, 21, 21, 21, 21, 21, 22, 23, 23, 23, 24, 24, 24

Offset: 1

Antti Karttunen, Nov 01 2012

nonn

Useful when computing A213719.

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

A230406 a(n) = A034968(A219666(n)); after zero, the differences between successive nodes in the infinite trunk of the factorial beanstalk (A219666).

Original entry on oeis.org

0, 1, 1, 3, 2, 3, 2, 5, 6, 2, 3, 2, 5, 5, 6, 2, 4, 5, 6, 7, 4, 5, 6, 7, 5, 5, 7, 10, 2, 3, 2, 5, 5, 6, 2, 4, 5, 6, 7, 4, 5, 6, 7, 5, 5, 6, 9, 8, 7, 10, 2, 4, 5, 6, 7, 4, 5, 6, 7, 5, 5, 6, 8, 6, 8, 8, 7, 10, 11, 4, 5, 6, 7, 5, 5, 6, 8, 6, 8, 8, 7, 10, 12, 10, 11

Offset: 0

Antti Karttunen, Nov 09 2013

nonn

Also the first differences of A219666, shifted once right and prepended with zero.

This sequence relates to the factorial base representation (A007623) in the same way as A213712 relates to the binary system.

Antti Karttunen, Table of n, a(n) for n = 0..3149

Cf. also A230418, A230410.

(define (A230406 n) (A034968 (A219666 n)))
;; Alternative definition:
(define (A230406 n) (if (zero? n) n (- (A219666 n) (A219666 (- n 1)))))

a(n) = A034968(A219666(n)).
a(0) = 0, and for n>=1, a(n) = A219666(n) - A219666(n-1).
a(A226061(n)) = A000217(n-1) for all n.

A255336 a(n) = A005811(A255056(n)); After the initial zero, the first differences of A255056.

Original entry on oeis.org

0, 2, 2, 2, 4, 2, 2, 4, 4, 4, 2, 2, 2, 4, 6, 4, 4, 4, 4, 2, 2, 2, 4, 6, 4, 6, 6, 4, 2, 4, 6, 4, 4, 4, 4, 2, 2, 2, 4, 6, 4, 6, 4, 4, 6, 6, 6, 6, 6, 4, 2, 4, 6, 4, 6, 6, 4, 2, 4, 6, 4, 4, 4, 4, 2, 2, 2, 4, 6, 4, 6, 4, 4, 6, 6, 6, 6, 4, 4, 6, 6, 6, 8, 6, 6, 6, 6, 6, 6, 4, 2, 4, 6, 4, 6, 4, 4, 6, 6, 6, 6, 6, 4, 2, 4, 6, 4, 6, 6, 4, 2, 4, 6, 4, 4, 4, 4, 2, 2

Offset: 0

Antti Karttunen, Feb 21 2015

nonn

First differences of A255056, shifted once right (prepended with zero).

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

First differences of A255056.
Terms of A255337 doubled.
Cf. A005811.
Analogous sequence: A213712.

a(n) = A005811(A255056(n)).
a(0) = 0; and for n >= 1: a(n) = A255056(n) - A255056(n-1).
a(n) = 2*A255337(n).

A255337 After a(0) = 0, the first differences of A255057: for n >= 1: a(n) = A255057(n) - A255057(n-1).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Feb 21 2015

nonn

Used for computing A255338 and A255339.

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

First differences of A255057.
Terms of A255336 divided by 2.
Cf. A005811, A213712, A255338, A255339.

a(n) = A005811(A255056(n))/2.
a(0) = 0; and for n >= 1: a(n) = A255057(n) - A255057(n-1).
a(n) = A255336(n)/2.

A278262 a(n) = A278222(A179016(n)).

Original entry on oeis.org

2, 4, 2, 8, 2, 12, 16, 2, 12, 24, 12, 32, 2, 12, 24, 30, 24, 12, 60, 24, 64, 2, 12, 24, 30, 24, 30, 210, 60, 48, 12, 60, 12, 180, 8, 24, 16, 128, 2, 12, 24, 30, 24, 30, 210, 60, 48, 30, 210, 30, 420, 12, 60, 24, 96, 12, 60, 12, 180, 60, 360, 360, 24, 216, 72, 16, 432, 256, 2, 12, 24, 30, 24, 30, 210, 60, 48, 30, 210, 30, 420, 12, 60, 24, 96, 30, 210, 30, 420

Offset: 1

Antti Karttunen, Nov 22 2016

Antti Karttunen, Table of n, a(n) for n = 1..16405
Index entries for sequences related to binary expansion of n

Cf. A179016, A278222.

Cf. also A213712, A278232.

(define (A278262 n) (A278222 (A179016 n)))

a(n) = A278222(A179016(n)).

A218541 First differences of A213715.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2, 1, 2, 2, 3, 1, 1, 2, 2, 2, 1, 2, 2, 3, 1, 2, 2, 2, 2, 2, 2, 3, 1, 1, 2, 2, 2, 1, 2, 2, 3, 1, 2, 2, 2, 2, 2, 2, 3, 1, 2, 2, 2, 2, 3, 3, 2, 3, 3, 2, 3, 4, 1, 1, 2, 2, 2, 1, 2, 2, 3, 1, 2, 2, 2, 2, 2, 2, 3, 1, 2

Offset: 0

Antti Karttunen, Nov 01 2012

nonn

This sequence seems to share similar fractal characteristics with A213712. See the given link to graph-plot.

Antti Karttunen, Table of n, a(n) for n = 0..8727
OEIS Server, Sequence plotted together with A213712 showing similar fractal features.

(define (A218541 n) (- (A213715 (1+ n)) (A213715 n)))

a(n) = A213715(n+1)-A213715(n).

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

cp's OEIS Frontend

A213718 n occurs A213712(n) times.

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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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A230406 a(n) = A034968(A219666(n)); after zero, the differences between successive nodes in the infinite trunk of the factorial beanstalk (A219666).

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A255336 a(n) = A005811(A255056(n)); After the initial zero, the first differences of A255056.

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A255337 After a(0) = 0, the first differences of A255057: for n >= 1: a(n) = A255057(n) - A255057(n-1).

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A278262 a(n) = A278222(A179016(n)).

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A218541 First differences of A213715.

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