cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A213709 Number of steps to go from 2^(n+1)-1 to (2^n)-1 using the iterative process described in A071542.

Original entry on oeis.org

1, 1, 2, 3, 5, 9, 17, 30, 54, 98, 179, 330, 614, 1150, 2162, 4072, 7678, 14496, 27418, 51979, 98800, 188309, 359889, 689649, 1325006, 2552031, 4926589, 9529551, 18463098, 35815293, 69534171, 135069124, 262448803, 510047416, 991381433, 1927317745, 3747885517
Offset: 0

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Author

Antti Karttunen, Oct 26 2012

Keywords

Comments

Also, apart from the first term a(0)=1, the number of terms in A179016 whose binary width is n+2 bits and whose second most significant bit is zero. For example, there is one term 4 (100) in three-bit range; two terms 8 (1000) and 11 (1011) in four bit range; three such terms: 16 (10000), 19 (10011) and 23 (10111) in five-bit range; five terms: 32, 35, 39, 42, 46 in six-bit range. This stems from the half-recursive nature of A179016, especially, that for all n >= 4, a(n) also gives the number of steps to go from (2^(n+1) + 2^n + 1) to 2^n using the iterative process described in A071542. Cf. also A226060. - Antti Karttunen, Jun 12 2013
Ratio a(n+1)/a(n) develops as: 1, 2, 1.5, 1.667..., 1.8, 1.889..., 1.765..., 1.8, 1.815..., 1.827..., 1.844..., 1.861..., 1.873..., 1.880..., 1.883..., 1.886..., 1.888..., 1.891..., 1.896..., 1.901..., 1.906..., 1.911..., 1.916..., 1.921..., 1.926..., 1.930..., 1.934..., 1.937..., 1.940..., 1.941..., 1.942..., 1.943..., 1.943..., 1.944..., 1.944..., 1.945..., 1.945..., 1.946..., 1.947..., 1.949..., 1.950..., 1.951..., 1.953..., 1.954..., 1.955..., 1.957..., 1.958... (which seem to converge slowly towards 2; see also comments at A218543).

Examples

			(2^0)-1 (0) is reached from (2^1)-1 (1) with one step by subtracting A000120(1) from 1.
(2^1)-1 (1) is reached from (2^2)-1 (3) with one step by subtracting A000120(3) from 3.
(2^2)-1 (3) is reached from (2^3)-1 (7) with two steps by first subtracting A000120(7) from 7 -> 4, and then subtracting A000120(4) from 4 -> 3.
Thus a(0)=a(1)=1 and a(2)=2.

Links

Crossrefs

First differences of A218600 and A213710. First differences of this sequence: A226060.
Cf. A179016, A213711.
Analogous sequence for factorial number system: A219661.

Formula

a(n) = A071542((2^(n+1))-1) - A071542((2^n)-1).
a(n) = A218542(n) + A218543(n) = A011782(n) - A213722(n).

Extensions

More terms from Antti Karttunen, Jun 05 2013

A233274 Relative offsets from the middle point of each row of A233271 & A218616 to the first point where the former exceeds the latter, which apart of case a(3)=-1 is always left of or at the middle point.

Original entry on oeis.org

0, 0, -1, 0, 0, 0, 0, 1, 2, 4, 8, 13, 22, 38, 68, 125, 232, 429, 786, 1428, 2578, 4645, 8364, 15064, 27145, 48990, 88736, 161813, 298001, 555451, 1048207, 1999608, 3844722, 7425094, 14356699, 27722560, 53374986
Offset: 1

Views

Author

Antti Karttunen, Jan 01 2014

Keywords

Comments

The sequence tells how many positions to the left of center of each row/subrange (of irregular tables like A233270, central point given by A233268) the sequences A233271 and A218616 cross each other (please see the linked graph).

Links

Crossrefs

Cf. A234020, A213709, A218600, A226060, A233268, A179016, A218616, A233270.

Programs

Formula

a(1)=a(2)=0, and for n > 2, a(n) = ⌈(A213709(n-1)/2)⌉ - A226060(n-2) - 1. Where ⌈x⌉ stands for ceiling(x)

A261235 First differences of A261234.

Original entry on oeis.org

1, 3, 7, 17, 45, 122, 334, 915, 2511, 6906, 19039, 52691, 146641, 410734, 1157321, 3276419, 9307640, 26509145, 75669108, 216483760, 620847130, 1784898316, 5143127022, 14847548759, 42923323624, 124213679008, 359730306331, 1042521363580, 3023562595587, 8776718908896, 25502164336020, 74179903571807, 216005883269160, 629644927525768, 1837129499003121, 5364782084798156
Offset: 0

Views

Author

Antti Karttunen, Aug 13 2015

Keywords

Crossrefs

Cf. A261234.
Cf. also A226060.

Programs

Formula

a(n) = A261234(n+1) - A261234(n).

Extensions

Terms from a(23) onward computed from the output of Hiroaki Yamanouchi's program (given in A261234) by Antti Karttunen, Aug 16 2015

A255069 First differences of A255071.

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 24, 44, 81, 150, 280, 526, 992, 1875, 3551, 6740, 12823, 24450, 46709, 89383, 171325, 328962, 632849, 1219909, 2356217, 4559224, 8835610, 17144046, 33295497, 64705083, 125802338, 244673791, 476011284, 926373373, 1803512210, 3512774806
Offset: 1

Views

Author

Antti Karttunen, Feb 21 2015

Keywords

Comments

Also, a(n) = the number of times a number whose binary expansion begins with 10... (cf. A004754) is encountered when iterating from 2^(n+2)-2 to (2^(n+1))-2 with the map x -> x - (number of runs in binary representation of x), i.e., with m(n) = A236840(n). For example, when starting from the initial value (2^(4+2))-2 = 62, we get m(62) = 60, m(60) = 58, m(58) = 54, m(54) = 50, m(50) = 46, m(46) = 42, m(42) = 36 and finally m(36) = 32, which is (2^(4+1)). Of the nine numbers encountered, only 46, 42, 36 and 32 (in binary: 101110, 101010, 100100 and 100000) are in A004754, thus a(4) = 5.

Crossrefs

First differences of A255071.
Cf. A004754, A005811, A236840, A255061, A255062, A255056, A255066.
Cf. also A255063, A255064, A255125, A255126.
Analogous sequence: A226060.

Programs

Formula

a(n) = A255071(n+1) - A255071(n).
For n > 1, a(n-1) = Sum_{k = A255062(n) .. A255061(n+1)}(1-secondmsb(A255056(k))).
Here secondmsb is implemented by the starting offset 2 version of A079944, and effectively gives the second most significant bit in the binary expansion of n. The formula follows from the semi-regular nature of number-of-runs beanstalk, see comments above and at A255071.
Showing 1-4 of 4 results.

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