A213709 -id:A213709 - cp's OEIS frontend

A233270 a(n) = A233271(n) - A179016(n).

0, 0, -1, 0, 0, 0, 1, 0, 0, 2, 1, 2, 0, 0, 3, 3, 4, 3, 4, 3, 3, 0, 0, 4, 4, 5, 4, 6, 5, 5, 3, 5, 5, 6, 4, 5, 4, 4, 0, 0, 5, 8, 9, 10, 13, 13, 15, 16, 17, 18, 18, 17, 17, 19, 19, 17, 17, 18, 18, 17, 16, 15, 13, 13, 10, 9, 8, 5, 0, 0, 6, 9, 14, 17, 18, 20, 22, 21

Offset: 0

Antti Karttunen, Dec 14 2013

For all n>=2, a(1+A213710(n)) = n-2.
Except for a(2)=-1 (which seems to be the only negative term in the sequence), the sequences A218600 and A213710 give the positions of zeros.
Furthermore, each subrange [A213710(n)..A218600(n+1)] is palindromic. A233268 gives the middle points of those ranges, the sequence A234018 gives the values at those points, while A234019 gives the maximum term in that range in this sequence.

			This irregular table begins as:
0;
0;
-1;
0, 0;
0, 1, 0;
0, 2, 1, 2, 0;
0, 3, 3, 4, 3, 4, 3, 3, 0;
0, 4, 4, 5, 4, 6, 5, 5, 3, 5, 5, 6, 4, 5, 4, 4, 0;
...
After zero, each row n is A213709(n-1) elements long.

Antti Karttunen, Rows 0..16, flattened

Except for a(2)=-1 (which seems to be the only negative term in the sequence), the sequences A218600 and A213710 give the positions of zeros.

Cf. A080468, A179016, A233271, A233268, A233274, A234018, A234019, A234020.

(define (A233270 n) (- (A233271 n) (A179016 n)))

a(n) = A233271(n) - A179016(n).

a(A218602(n)) = a(n). [This is just a claim that each row is palindrome]

A218542 Number of times when an even number is encountered, when going from 2^(n+1)-1 to (2^n)-1 using the iterative process described in A071542.

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 8, 12, 23, 44, 86, 163, 308, 576, 1074, 1991, 3680, 6800, 12626, 23644, 44751, 85567, 164941, 319694, 621671, 1211197, 2362808, 4614173, 9018299, 17635055, 34486330, 67408501, 131642673, 256795173, 500346954, 973913365, 1894371802, 3683559071

Offset: 0

Antti Karttunen, Nov 02 2012

nonn

Ratio a(n)/A213709(n) develops as: 1, 0, 0.5, 0.333..., 0.4, 0.333..., 0.471..., 0.400..., 0.426..., 0.449..., 0.480..., 0.494..., 0.502..., 0.501..., 0.497..., 0.489..., 0.479..., 0.469..., 0.461..., 0.455..., 0.453..., 0.454..., 0.458..., 0.464..., 0.469..., 0.475..., 0.480..., 0.484..., 0.488..., 0.492..., 0.496..., 0.499..., 0.502..., 0.503..., 0.505..., 0.505..., 0.505..., 0.505..., 0.505..., 0.504..., 0.504..., 0.503..., 0.503..., 0.502..., 0.502..., 0.502..., 0.503..., 0.503... (See further comments at A218543).

			(2^0)-1 (0) is reached from (2^1)-1 (1) with one step by subtracting A000120(1) from 1. Zero is an even number, so a(0)=1.
(2^1)-1 (1) is reached from (2^2)-1 (3) with one step by subtracting A000120(3) from 3. One is not an even number, so a(1)=0.
(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. Four is an even number, but three is not, so a(2)=1.

Antti Karttunen, Table of n, a(n) for n = 0..47

Cf. A219662 (analogous sequence for factorial number system).

a(n) = Sum_{i=A218600(n) .. (A218600(n+1)-1)} A213728(i).

a(n) = A213709(n) - A218543(n).

More terms from Antti Karttunen, Jun 05 2013

A218543 Number of times when an odd number is encountered, when going from 2^(n+1)-1 to (2^n)-1 using the iterative process described in A071542.

Original entry on oeis.org

0, 1, 1, 2, 3, 6, 9, 18, 31, 54, 93, 167, 306, 574, 1088, 2081, 3998, 7696, 14792, 28335, 54049, 102742, 194948, 369955, 703335, 1340834, 2563781, 4915378, 9444799, 18180238, 35047841, 67660623, 130806130, 253252243, 491034479, 953404380, 1853513715, 3607440034

Offset: 0

Antti Karttunen, Nov 02 2012

nonn

Ratio a(n)/A213709(n) develops as: 0, 1, 0.5, 0.666..., 0.6, 0.666..., 0.529..., 0.6, 0.574..., 0.551..., 0.520..., 0.506..., 0.498..., 0.499..., 0.503..., 0.511..., 0.521..., 0.531..., 0.539..., 0.545..., 0.547..., 0.546..., 0.542..., 0.536..., 0.531..., 0.525..., 0.520..., 0.516..., 0.512..., 0.508..., 0.504..., 0.501..., 0.498..., 0.497..., 0.495..., 0.495..., 0.495..., 0.495..., 0.495..., 0.496..., 0.496..., 0.497..., 0.497..., 0.498..., 0.498..., 0.498..., 0.497..., 0.497...
Ratio a(n)/A218542(n) develops as follows from n>=2 onward:
1, 2, 1.5, 2, 1.125, 1.5, 1.348..., 1.227..., 1.081..., 1.025..., 0.994..., 0.997..., 1.013..., 1.045..., 1.086..., 1.132..., 1.172..., 1.198..., 1.208..., 1.201..., 1.182..., 1.157..., 1.131..., 1.107..., 1.085..., 1.065..., 1.047..., 1.031..., 1.016..., 1.004..., 0.994..., 0.986..., 0.981..., 0.979..., 0.978..., 0.979..., 0.981..., 0.983..., 0.986..., 0.988..., 0.989..., 0.990..., 0.991..., 0.991..., 0.989..., 0.987...
Observation: A179016 seems to alternatively slightly favor the odd numbers and then again the even numbers, at least for the terms computed so far.
Please plot this sequence against A218542 in the "ratio mode" (given as a link) to see how smoothly (almost "continuously") the ratios given above develop.
What is the reason for that smoothness? (Conjecture: The distribution of "tendrils", i.e. finite subtrees in the beanstalk and its almost fractal nature? Cf: A218787.)

			(2^0)-1 (0) is reached from (2^1)-1 (1) with one step by subtracting A000120(1) from 1. Zero is not an odd number, so a(0)=0.
(2^1)-1 (1) is reached from (2^2)-1 (3) with one step by subtracting A000120(3) from 3. One is an odd number, so a(1)=1.
(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. Four is not an odd number, but three is, so a(2)=1.

Antti Karttunen, Table of n, a(n) for n = 0..47
OEIS Server, Sequence plotted together with A218542 showing how their ratio develops.

a(n) = A213709(n)-A218542(n). Cf. A213733, A218787, A218789.

Analogous sequence for factorial number system: A219663.

a(n) = Sum_{i=A218600(n) .. (A218600(n+1)-1)} A213729(i)

A218616 The infinite trunk of beanstalk (A179016) with reversed subsections.

Original entry on oeis.org

0, 1, 3, 7, 4, 15, 11, 8, 31, 26, 23, 19, 16, 63, 57, 53, 49, 46, 42, 39, 35, 32, 127, 120, 116, 112, 109, 104, 101, 97, 94, 89, 85, 81, 78, 74, 71, 67, 64, 255, 247, 240, 236, 231, 225, 221, 215, 209, 205, 200, 197, 193, 190, 184, 180, 176, 173, 168, 165, 161

Offset: 0

Antti Karttunen, Nov 10 2012

This can be viewed as an irregular table: after the initial zero on row 0, start each row n with (2^n)-1 and subtract repeatedly the number of 1-bits to get successive terms, until the number that has already been listed (which is always (2^(n-1))-1) is encountered, which is not listed second time, but instead, the current row is finished and the next row starts with (2^(n+1))-1, with the same process repeated.
This contains the terms in the infinite trunk of beanstalk (A179016) listed in partially reversed manner: after the initial zero each subsequence lists A213709(n) successive terms  from A179016, descending from (2^n)-1 downwards, usually down to 2^(n-1) (conjectured to indeed be a power of 2 in each case, apart from 2 itself missing from the beginning of the sequence).
Currently A179016 and many of the derived sequences are much easier and somewhat faster to compute with the help of this sequence, especially if the program computes any other required values incrementally in the same loop.

			After zero, we start with (2^1)-1 = 1, subtract A000120(1)=1 from it, resulting 1-1=0 (which is of the form (2^0)-1, thus not listed second time), instead, start the next row with (2^2)-1 = 3, subtract A000120(3)=2 from it, resulting 3-2=1, which has been already encountered, thus start the next row with (2^3)-1 = 7, subtract A000120(7)=3 from it, resulting 7-3=4, which is listed after 7, then 4-A000120(4)=4-1=3, which is of the form (2^k)-1 and already encountered, thus start the next row with (2^4)-1 = 15, etc. This results an irregular table which begins as:
0; 1; 3; 7, 4; 15, 11, 8; 31, 26, 23, 19, 16; 63, 57, ...
After zero, each row n is A213709(n-1) elements long.

Antti Karttunen, Rows 0..16, flattened

a(n) = A179016(A218602(n)). The rows are the initial portions of every (2^n)-1:th row in A218254.

a(0)=0, a(1)=1, and for n > 1, if n = A213710(A213711(n)-1) then a(n) = (2^A213711(n)) - 1, and in other cases, a(n) = A011371(a(n-1)).

Alternatively: For n < 4, a(n) = (2^n)-1, and for n >= 4, a(n) = A004755(A004755(A011371(a(n-1)))) if A011371(a(n-1))+1 is power of 2, otherwise just A011371(a(n-1)).

A261234 a(n) = number of steps to reach (3^n)-1 when starting from k = (3^(n+1))-1 and repeatedly applying the map that replaces k with k - (sum of digits in base-3 representation of k).

Original entry on oeis.org

1, 2, 5, 12, 29, 74, 196, 530, 1445, 3956, 10862, 29901, 82592, 229233, 639967, 1797288, 5073707, 14381347, 40890492, 116559600, 333043360, 953890490, 2738788806, 7881915828, 22729464587, 65652788211, 189866467219, 549596773550, 1592118137130, 4615680732717, 13392399641613, 38894563977633, 113074467549440, 329080350818600, 958725278344368, 2795854777347489

Offset: 0

Antti Karttunen, Aug 13 2015

Hiroaki Yamanouchi, Table of n, a(n) for n = 0..100
Antti Karttunen, Naive C-program for computing the terms of A261234, A261236 and A261237 at the same time
Hiroaki Yamanouchi, Fast Python-program for computing terms of A213709, A261234 and analogous sequences in other bases

Cf. A000244, A054861, A261231, A261230.
First differences of A261232 and A261233.
Sum of A261236 and A261237.
Cf. A261235 (first differences of this sequence).
Cf. also A213709.

Table[Length@ NestWhileList[# - Total@ IntegerDigits[#, 3] &, 3^(n + 1) - 1, # > 3^n - 1 &] - 1, {n, 0, 16}] (* Michael De Vlieger, Jun 27 2016 *)

a(n) = A261236(n) + A261237(n).

a(23)-a(35) from Hiroaki Yamanouchi, Aug 16 2015

A213711 a(n) = minimal k for which A218600(k) >= n.

Original entry on oeis.org

0, 1, 2, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9

Offset: 0

Antti Karttunen, Oct 26 2012

nonn

a(0)=0, and after that each term n occurs A213709(n-1) times.

Auxiliary function for computing A179016.

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

Cf. A179016, A213709, A218600.

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

A233268 The middle point of row n in binary beanstalk related sequences A179016, A218602, A218616, A233270, A233271.

Original entry on oeis.org

1, 2, 3, 6, 10, 17, 30, 53, 95, 171, 310, 564, 1036, 1918, 3574, 6691, 12566, 23653, 44610, 84309, 159698, 303253, 577352, 1102121, 2109448, 4047967, 7787277, 15015347, 29011671, 56150867, 108825599, 211127246, 409886210, 796134319, 1546848744, 3006198333, 5843799964

Offset: 1

Antti Karttunen, Dec 29 2013

nonn

a(n) points to the center of each palindromic row/subrange of A233270, and to the lower position nearest to the center, if the length of range is even.

For all n, A218602(a(n)) = a(n) + (1-A000035(A213709(n-1))).

Cf. A213709, A213710, A218600, A234018, A234019, A234020.

Cf. also A179016, A218602, A218616, A233270, A233271.

a(n) = floor((A213710(n-1) + A218600(n)) / 2).

a(n) = A218600(n-1) + ceiling((A213709(n-1)/2)).

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

Antti Karttunen, Jan 01 2014

sign

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).

A. Karttunen, Sequences A233271 and A218616 compared with OEIS Plot2-script

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

(define (A233274 n) (if (< n 3) 0 (- (+ -1 (ceiling->exact (/ (A213709 (- n 1)) 2))) (A226060 (- n 2)))))
(define (A233274v2 n) (if (< n 2) 0 (- (A233268 n) (+ (A218600 (- n 1)) (A226060 (- n 2))) 1)))

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)

A257259 a(n) = A218542(n) - A218543(n).

Original entry on oeis.org

1, -1, 0, -1, -1, -3, -1, -6, -8, -10, -7, -4, 2, 2, -14, -90, -318, -896, -2166, -4691, -9298, -17175, -30007, -50261, -81664, -129637, -200973, -301205, -426500, -545183, -561511, -252122, 836543, 3542930, 9312475, 20508985, 40858087, 76119037, 135203839, 232236417, 390709345, 652711050, 1101492173, 1921013671, 3558812217, 7170401227, 15800043812, 37400877416

Offset: 0

Antti Karttunen, May 13 2015

sign

a(n) = the difference between the number of even and odd numbers encountered when traversing from 2^(n+1)-1 to (2^n)-1 by iterating the map A011371: x -> x - (number of 1's in binary representation of x).

Cf. A011371, A218542, A218543, A213709.

Partial sums: A257805.

(define (A257259 n) (- (A218542 n) (A218543 n)))

a(n) = A218542(n) - A218543(n).

A213722 a(n) = number of terms in A213717 whose magnitude is in range [(2^n)-1,(2^(n+1))-2] (or equally, in range [(2^n),(2^(n+1))-1]).

Original entry on oeis.org

0, 0, 0, 1, 3, 7, 15, 34, 74, 158, 333, 694, 1434, 2946, 6030, 12312, 25090, 51040, 103654, 210165, 425488, 860267, 1737263, 3504655, 7063602, 14225185, 28627843, 57579313

Offset: 0

Antti Karttunen, Nov 01 2012

nonn

a(n) = A011782(n)-A213709(n).

a(n) = Sum_{i=(2^n) .. (2^(n+1))-1} (A079559(i)-A213719(i)).

cp's OEIS Frontend

A233270 a(n) = A233271(n) - A179016(n).

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A218542 Number of times when an even number is encountered, when going from 2^(n+1)-1 to (2^n)-1 using the iterative process described in A071542.

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A218543 Number of times when an odd number is encountered, when going from 2^(n+1)-1 to (2^n)-1 using the iterative process described in A071542.

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A218616 The infinite trunk of beanstalk (A179016) with reversed subsections.

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A261234 a(n) = number of steps to reach (3^n)-1 when starting from k = (3^(n+1))-1 and repeatedly applying the map that replaces k with k - (sum of digits in base-3 representation of k).

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Mathematica

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A213711 a(n) = minimal k for which A218600(k) >= n.

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A233268 The middle point of row n in binary beanstalk related sequences A179016, A218602, A218616, A233270, A233271.

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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.

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A257259 a(n) = A218542(n) - A218543(n).

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