A318921 -id:A318921 - cp's OEIS frontend

A318928 Runs-resistance of binary representation of n.

1, 2, 1, 3, 2, 3, 1, 3, 3, 2, 4, 2, 4, 3, 1, 3, 3, 5, 4, 4, 2, 5, 4, 3, 4, 4, 3, 3, 4, 3, 1, 3, 3, 5, 3, 3, 5, 4, 3, 4, 5, 2, 4, 3, 4, 5, 4, 3, 3, 3, 2, 4, 4, 3, 3, 2, 3, 4, 3, 3, 4, 3, 1, 3, 3, 5, 3, 3, 5, 3, 4, 3, 3, 5, 6, 4, 5, 3, 3, 4, 5, 4, 4, 4, 2, 5, 4, 5, 5, 4, 5, 5, 4, 5, 4

Offset: 1

N. J. A. Sloane, Sep 09 2018

Following Lenormand (2003), we define the "runs-resistance" of a finite list L to be the number of times the RUNS transformation must be applied to L in order to reduce L to a list with a single element.
Here it is immaterial whether we read the binary representation of n from left to right or right to left.
The RUNS transformation must be applied at least once, in order to obtain a list, so a(n) >= 1.

			11 in binary is [1, 0, 1, 1],
which has runs of lengths [1, 1, 2],
which has runs of lengths [2, 1],
which has runs of lengths [1, 1],
which has a single run of length [2].
This took four steps, so a(11) = 4.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000
Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003. Apparently unpublished. This is a scanned copy of the version that the author sent to me in 2003.
N. J. A. Sloane, Transforms

See A319103 for an inverse, and A319417 and A319418 for records.
Ignoring the first digit gives A329870.
Cuts-resistance is A319416.
Compositions counted by runs-resistance are A329744.
Binary words counted by runs-resistance are A319411 and A329767.
Cf. A107907, A319420, A329860, A329861, A329865, A329867.

with(transforms);
# compute Lenormand's "resistance" of a list
resist:=proc(a) local ct,i,b;
if whattype(a) <> list then ERROR("input must be a list"); fi:
ct:=0; b:=a; for i from 1 to 100 do
if nops(b)=1 then return(ct); fi;
b:=RUNS(b); ct:=ct+1; od; end;
a:=[1];
for n from 2 to 100 do
b:=convert(n,base,2);
r:=resist(b);
a:=[op(a),r];
od:

Table[If[n == 1, 1, Length[NestWhileList[Length/@Split[#] &, IntegerDigits[n, 2], Length[#] > 1 &]] - 1], {n, 50}] (* Gus Wiseman, Nov 25 2019 *)

a(1) corrected by N. J. A. Sloane, Sep 20 2018

A333627 The a(n)-th composition in standard order is the sequence of run-lengths of the n-th composition in standard order.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 3, 4, 1, 3, 2, 6, 3, 7, 5, 8, 1, 3, 3, 6, 3, 5, 7, 12, 3, 7, 6, 14, 5, 11, 9, 16, 1, 3, 3, 6, 2, 7, 7, 12, 3, 7, 4, 10, 7, 15, 13, 24, 3, 7, 7, 14, 7, 13, 15, 28, 5, 11, 10, 22, 9, 19, 17, 32, 1, 3, 3, 6, 3, 7, 7, 12, 3, 5, 6, 14, 7, 15, 13

Offset: 0

Gus Wiseman, Mar 30 2020

nonn

A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The standard compositions and their run-lengths:
       0 ~ () -> () ~ 0
      1 ~ (1) -> (1) ~ 1
      2 ~ (2) -> (1) ~ 1
     3 ~ (11) -> (2) ~ 2
      4 ~ (3) -> (1) ~ 1
     5 ~ (21) -> (11) ~ 3
     6 ~ (12) -> (11) ~ 3
    7 ~ (111) -> (3) ~ 4
      8 ~ (4) -> (1) ~ 1
     9 ~ (31) -> (11) ~ 3
    10 ~ (22) -> (2) ~ 2
   11 ~ (211) -> (12) ~ 6
    12 ~ (13) -> (11) ~ 3
   13 ~ (121) -> (111) ~ 7
   14 ~ (112) -> (21) ~ 5
  15 ~ (1111) -> (4) ~ 8
     16 ~ (5) -> (1) ~ 1
    17 ~ (41) -> (11) ~ 3
    18 ~ (32) -> (11) ~ 3
   19 ~ (311) -> (12) ~ 6

Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003.

Positions of first appearances are A333630.
All of the following pertain to compositions in standard order (A066099):
- The length is A000120.
- The partial sums from the right are A048793.
- The sum is A070939.
- Adjacent equal pairs are counted by A124762.
- Equal runs are counted by A124767.
- Strict compositions are ranked by A233564.
- The partial sums from the left are A272020.
- Constant compositions are ranked by A272919.
- Normal compositions are ranked by A333217.
- Heinz number is A333219.
- Anti-runs are counted by A333381.
- Adjacent unequal pairs are counted by A333382.
- Runs-resistance is A333628.
- First appearances of run-resistances are A333629.
Cf. A029931, A098504, A114994, A225620, A228351, A238279, A242882, A318928, A329744, A329747, A333489.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Total[2^(Accumulate[Reverse[Length/@Split[stc[n]]]])]/2,{n,0,30}]

A000120(n) = A070939(a(n)).

A000120(a(n)) = A124767(n).

A329744 Triangle read by rows where T(n,k) is the number of compositions of n > 0 with runs-resistance k, 0 <= k <= n - 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 3, 2, 1, 1, 6, 6, 2, 1, 3, 15, 9, 4, 0, 1, 1, 22, 22, 16, 2, 0, 1, 3, 41, 38, 37, 8, 0, 0, 1, 2, 72, 69, 86, 26, 0, 0, 0, 1, 3, 129, 124, 175, 78, 2, 0, 0, 0, 1, 1, 213, 226, 367, 202, 14, 0, 0, 0, 0, 1, 5, 395, 376, 750, 469, 52, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Nov 21 2019

A composition of n is a finite sequence of positive integers with sum n.

For the operation of taking the sequence of run-lengths of a finite sequence, runs-resistance is defined as the number of applications required to reach a singleton.

			Triangle begins:
   1
   1   1
   1   1   2
   1   2   3   2
   1   1   6   6   2
   1   3  15   9   4   0
   1   1  22  22  16   2   0
   1   3  41  38  37   8   0   0
   1   2  72  69  86  26   0   0   0
   1   3 129 124 175  78   2   0   0   0
   1   1 213 226 367 202  14   0   0   0   0
   1   5 395 376 750 469  52   0   0   0   0   0
Row n = 6 counts the following compositions:
  (6)  (33)      (15)    (114)    (1131)
       (222)     (24)    (411)    (1311)
       (111111)  (42)    (1113)   (11121)
                 (51)    (1221)   (12111)
                 (123)   (2112)
                 (132)   (3111)
                 (141)   (11112)
                 (213)   (11211)
                 (231)   (21111)
                 (312)
                 (321)
                 (1122)
                 (1212)
                 (2121)
                 (2211)

Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003.

Row sums are A000079.
Column k = 1 is A032741.
Column k = 2 is A329745.
Column k = n - 2 is A329743.
The version for partitions is A329746.
The version with rows reversed is A329750.
Cf. A000740, A008965, A098504, A242882, A318928, A329747.

runsres[q_]:=Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1;
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],runsres[#]==k&]],{n,10},{k,0,n-1}]

A027649 a(n) = 2*(3^n) - 2^n.

Original entry on oeis.org

1, 4, 14, 46, 146, 454, 1394, 4246, 12866, 38854, 117074, 352246, 1058786, 3180454, 9549554, 28665046, 86027906, 258149254, 774578834, 2323998646, 6972520226, 20918609254, 62757924914, 188277969046, 564842295746, 1694543664454, 5083664547794, 15251060752246

Offset: 0

N. J. A. Sloane

Poly-Bernoulli numbers B_n^(k) with k=-2.
Binomial transform of A007051, if both sequences start at 0. Binomial transform of A000225(n+1). - Paul Barry, Mar 24 2003
Euler expands (1-z)/(1-5z+6z^2) and finds the general term. Section 226 of the Introductio indicates that he could have written down the recursion relation: a(n) = 5 a(n-1)-6 a(n-2). - V. Frederick Rickey (fred-rickey(AT)usma.edu), Feb 10 2006
Let R be a binary relation on the power set P(A) of a set A having n = |A| elements such that for every element x, y of P(A), xRy if x is a subset of y or y is a subset of x. Then a(n) = |R|. - Ross La Haye, Dec 22 2006
With regard to the comment by Ross La Haye: For proper subsets see A056182. - For nonempty subsets see A091344. - For nonempty proper subsets see a(n+1) in A260217. - Manfred Boergens, Aug 02 2023
If x, y are two n-bit binary strings then a(n) gives the number of pairs (x,y) such that XOR(x, y) = ABS(x - y). - Ramasamy Chandramouli, Feb 15 2009
Equals row sums of the triangular version of A038573. - Gary W. Adamson, Jun 04 2009
Inverse binomial transform of A085350. - Paul Curtz, Nov 14 2009
Related to the number of even a's in a nontrivial cycle (should one exist) in the 3x+1 Problem, where a <= floor(log_2(2*(3^n) - 2^n)). The value n correlates to the number of odds in such a nontrivial cycle. See page 1288 of Crandall's paper. Also, this relation gives another proof that the number of odds divided by the number of evens in a nontrivial cycle is bounded by log 2 / log 3 (this observation does not resolve the finite cycles conjecture as the value could be arbitrarily close to this bound). However, the same argument gives that log 2 / log 3 is less than or equal to the number of odds divided by the number of evens in a divergent sequence (should one exist), as log 2 / log 3 is the limit value for a cycle of an arbitrarily large length, where the length is given by the value n. - Jeffrey R. Goodwin, Aug 04 2011
Row sums of Riordan triangle A106516. - Wolfdieter Lang, Jan 09 2015
Number of restricted barred preferential arrangements having 3 bars in which the sections are all restricted sections such that (for fixed sections i and j) section i or section j is empty. - Sithembele Nkonkobe, Oct 12 2015
This is also row 2 of A281891: for n >= 1, when consecutive positive integers are written as a product of primes in nondecreasing order, a factor of 2 or 3 occurs in n-th position a(n) times out of every 6^n. - Peter Munn, May 18 2017
Also row sums of A124929. - Omar E. Pol, Jun 15 2017
This is the sum of A318921(n) for n in the range 2^(k+1) to 2^(k+2)-1. See A318921 for proof. - N. J. A. Sloane, Sep 25 2018
a(n) is also the number of acyclic orientations of the complete bipartite graph K_{2,n}. - Vincent Pilaud, Sep 15 2020
a(n-1) is also the number of n-digit numbers whose largest decimal digit is 2. - Stefano Spezia, Nov 15 2023

Leonhard Euler, Introductio in analysin infinitorum (1748), section 216.

T. D. Noe, Table of n, a(n) for n = 0..200
Taylor Brysiewicz, Holger Eble, and Lukas Kühne, Enumerating chambers of hyperplane arrangements with symmetry, arXiv:2105.14542 [math.CO], 2021.
R. E. Crandall, On the 3x+1 problem, Math. Comp., 32 (1978) 1281-1292.
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata, arXiv:1503.01796 [math.CO], 2015; see also the Accompanying Maple Package.
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, Odd-Rule Cellular Automata on the Square Grid, arXiv:1503.04249 [math.CO], 2015.
John Elias, Illustration of initial terms: Reflected Sierpinski triangle
K. Imatomi, M. Kaneko, and E. Takeda, Multi-Poly-Bernoulli Numbers and Finite Multiple Zeta Values, J. Int. Seq. 17 (2014) # 14.4.5
Ken Kamano, Sums of Products of Poly-Bernoulli Numbers of Negative Index, Journal of Integer Sequences, Vol. 15 (2012), #12.1.3.
K. Kamano, Sums of Products of Bernoulli Numbers, Including Poly-Bernoulli Numbers, J. Int. Seq. 13 (2010), 10.5.2.
Masanobu Kaneko, Poly-Bernoulli numbers, Journal de théorie des nombres de Bordeaux, 9 no. 1 (1997), Pages 221-228.
Takao Komatsu, Some recurrence relations of poly-Cauchy numbers, J. Nonlinear Sci. Appl., (2019) Vol. 12, Issue 12, 829-845.
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
S. Nkonkobe and V. Murali, On some properties and relations between restricted barred preferential arrangements, multi-poly-bernoulli numbers and related numbers, arXiv:1509.07352 [math.CO], 2015.
Eric Weisstein's World of Mathematics, Sierpinski Sieve
Wikipedia, Sierpinski triangle
Index entries for sequences related to Bernoulli numbers.
Index entries for linear recurrences with constant coefficients, signature (5,-6).

Row n = 2 of array A099594.
Also occurs as a row, column, diagonal or as row sums in A038573, A085870, A090888, A106516, A217764, A281891.
Cf. A000079, A000225, A001047, A007051, A053581, A056182, A085350, A091344, A124929, A166060, A260217, A318921.

a027649 n = a027649_list !! n
a027649_list = map fst $ iterate (\(u, v) -> (3 * u + v, 2 * v)) (1, 1)
-- Reinhard Zumkeller, Jun 09 2013

[2*(3^n)-2^n: n in [0..30]]; // Vincenzo Librandi, Jul 17 2011

a(n, k):= (-1)^n*sum( (-1)^'m'*'m'!*Stirling2(n,'m')/('m'+1)^k,'m'=0..n);
seq(a(n, -2), n=0..30);

Table[2(3^n)-2^n,{n,0,30}] (* or *) LinearRecurrence[ {5,-6},{1,4},31]  (* Harvey P. Dale, Apr 22 2011 *)

a(n)=2*(3^n)-2^n \\ Charles R Greathouse IV, Jul 16 2011

Vec((1-x)/((1-2*x)*(1-3*x)) + O(x^50)) \\ Altug Alkan, Oct 12 2015

[2*(3^n - 2^(n-1)) for n in (0..30)] # G. C. Greubel, Aug 01 2022

G.f.: (1-x)/((1-2*x)*(1-3*x)).
a(n) = 3*a(n-1) + 2^(n-1), with a(0) = 1.
a(n) = Sum_{k=0..n} binomial(n, k)*(2^(k+1) - 1). - Paul Barry, Mar 24 2003
Partial sums of A053581. - Paul Barry, Jun 26 2003
Main diagonal of array (A085870) defined by T(i, 1) = 2^i - 1, T(1, j) = 2^j - 1, T(i, j) = T(i-1, j) + T(i-1, j-1). - Benoit Cloitre, Aug 05 2003
a(n) = A090888(n, 3). - Ross La Haye, Sep 21 2004
a(n) = Sum_{k=0..n} binomial(n+2, k+1)*Sum_{j=0..floor(k/2)} A001045(k-2j). - Paul Barry, Apr 17 2005
a(n) = Sum_{k=0..n} Sum_{j=0..n} binomial(n,j)*binomial(j+1,k+1). - Paul Barry, Sep 18 2006
a(n) = A166060(n+1)/6. - Philippe Deléham, Oct 21 2009
a(n) = 5*a(n-1) - 6*a(n-2), a(0)=1, a(1)=4. - Harvey P. Dale, Apr 22 2011
a(n) = A217764(n,2). - Ross La Haye, Mar 27 2013
For n>0, a(n) = 3 * a(n-1) + 2^(n-1) = 2 * (a(n-1) + 3^(n-1)). - J. Conrad, Oct 29 2015
for n>0, a(n) = 2 * (1 + 2^(n-2) + Sum_{x=1..n-2} Sum_{k=0..x-1} (binomial(x-1,k)*(2^(k+1) + 2^(n-x+k)))). - J. Conrad, Dec 10 2015
E.g.f.: exp(2*x)*(2*exp(x) - 1). - Stefano Spezia, May 18 2024

Better formulas from David W. Wilson and Michael Somos
Incorrect formula removed by Charles R Greathouse IV, Mar 18 2010
Duplications (due to corrections to A numbers) removed by Peter Munn, Jun 15 2017

A333628 Runs-resistance of the n-th composition in standard order. Number of steps taking run-lengths to reduce the n-th composition in standard order to a singleton.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Mar 31 2020

nonn

For the operation of taking the sequence of run-lengths of a finite sequence, runs-resistance is defined as the number of applications required to reach a singleton.

			Starting with 13789 and repeatedly applying A333627 gives: 13789 -> 859 -> 110 -> 29 -> 11 -> 6 -> 3 -> 2, corresponding to the compositions: (1,2,2,1,1,2,1,1,2,1) -> (1,2,2,1,2,1,1) -> (1,2,1,1,2) -> (1,1,2,1) -> (2,1,1) -> (1,2) -> (1,1) -> (2), so a(13789) = 7.

Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003.

Number of times applying A333627 to reach a power of 2, starting with n.
Positions of first appearances are A333629.
All of the following pertain to compositions in standard order (A066099):
- The length is A000120.
- The partial sums from the right are A048793.
- The sum is A070939.
- Adjacent equal pairs are counted by A124762.
- Equal runs are counted by A124767.
- Strict compositions are ranked by A233564.
- The partial sums from the left are A272020.
- Constant compositions are ranked by A272919.
- Normal compositions are ranked by A333217.
- Heinz number is A333219.
- Anti-runs are counted by A333381.
- Adjacent unequal pairs are counted by A333382.
- First appearances for specified run-lengths are A333630.
Cf. A029931, A098504, A114994, A225620, A228351, A238279, A242882, A318928, A329744, A329747, A333489.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
runsres[q_]:=Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1;
Table[runsres[stc[n]],{n,100}]

A329746 Triangle read by rows where T(n,k) is the number of integer partitions of n > 0 with runs-resistance k, 0 <= k <= n - 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 0, 1, 3, 4, 3, 0, 0, 1, 1, 4, 8, 1, 0, 0, 1, 3, 6, 10, 2, 0, 0, 0, 1, 2, 8, 13, 6, 0, 0, 0, 0, 1, 3, 11, 20, 7, 0, 0, 0, 0, 0, 1, 1, 11, 29, 14, 0, 0, 0, 0, 0, 0, 1, 5, 19, 31, 20, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Nov 21 2019

For the operation of taking the sequence of run-lengths of a finite sequence, runs-resistance is defined as the number of applications required to reach a singleton.

			Triangle begins:
  1
  1  1
  1  1  1
  1  2  1  1
  1  1  2  3  0
  1  3  4  3  0  0
  1  1  4  8  1  0  0
  1  3  6 10  2  0  0  0
  1  2  8 13  6  0  0  0  0
  1  3 11 20  7  0  0  0  0  0
  1  1 11 29 14  0  0  0  0  0  0
  1  5 19 31 20  1  0  0  0  0  0  0
  1  1 17 50 30  2  0  0  0  0  0  0  0
  1  3 25 64 37  5  0  0  0  0  0  0  0  0
  1  3 29 74 62  7  0  0  0  0  0  0  0  0  0
Row n = 8 counts the following partitions:
  (8)  (44)        (53)    (332)      (4211)
       (2222)      (62)    (422)      (32111)
       (11111111)  (71)    (611)
                   (431)   (3221)
                   (521)   (5111)
                   (3311)  (22211)
                           (41111)
                           (221111)
                           (311111)
                           (2111111)

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)
Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003.

Row sums are A000041.
Column k = 1 is A032741.
Column k = 2 is A329745.
A similar invariant is frequency depth; see A323014, A325280.
The version for compositions is A329744.
The version for binary words is A329767.
Cf. A098504, A182850, A225485, A242882, A318928, A325410, A329747.

runsres[q_]:=Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1;
Table[Length[Select[IntegerPartitions[n],runsres[#]==k&]],{n,10},{k,0,n-1}]

\\ rr(p) gives runs resistance of partition.
rr(p)={my(r=0); while(#p > 1, my(L=List(), k=0); for(i=1, #p, if(i==#p||p[i]<>p[i+1], listput(L, i-k); k=i)); p=Vec(L); r++); r}
row(n)={my(v=vector(n)); forpart(p=n, v[1+rr(Vec(p))]++); v}
{ for(n=1, 10, print(row(n))) } \\ Andrew Howroyd, Jan 19 2023

cp's OEIS Frontend

A318923 Apply Lenormand's transformation k -> A318921(k) to the primes.

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A318925 a(n) = A175046(A318921(n)).

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A319409 a(n) = n - A318921(n).

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A318922 Apply Lenormand's transformation k -> A318921(k) to the Fibonacci numbers.

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A318928 Runs-resistance of binary representation of n.

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A333627 The a(n)-th composition in standard order is the sequence of run-lengths of the n-th composition in standard order.

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A329744 Triangle read by rows where T(n,k) is the number of compositions of n > 0 with runs-resistance k, 0 <= k <= n - 1.

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Mathematica

A027649 a(n) = 2*(3^n) - 2^n.

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