A114901 -id:A114901 - cp's OEIS frontend

A037201 Differences between consecutive primes (A001223) but with repeats omitted.

1, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 4, 6, 2, 10, 2, 4, 2, 12, 4, 2, 4, 6, 2, 10, 6, 2, 6, 4, 2, 10, 14, 4, 2, 4, 14, 6, 10, 2, 4, 6, 8, 6, 4, 6, 8, 4, 8, 10, 2, 10, 2, 6, 4, 6, 8, 4

Offset: 1

Naohiro Nomoto

Also the run-compression of the sequence of first differences of prime numbers, where we define the run-compression of a sequence to be the anti-run obtained by reducing each run of repeated parts to a single part. Alternatively, we can remove all parts equal to the part immediately to their left. For example, (1,1,2,2,1) has run-compression (1,2,1). - Gus Wiseman, Sep 16 2024

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for primes, gaps between

This is the run-compression of A001223 = first differences of A000040.
The repeats were at positions A064113 before being omitted.
Adding up runs instead of compressing them gives A373822.
The even terms halved are A373947.
For prime-powers instead of prime numbers we have A376308.
Positions of first appearances are A376520, sorted A376521.
A003242 counts compressed compositions.
A333254 lists run-lengths of differences between consecutive primes.
A373948 encodes compression using compositions in standard order.
Cf. A007921, A030173, A053289, A106356, A114901, A116608, A116861, A124767, A238130, A333755, A335406, A373954.

a037201 n = a037201_list !! (n-1)
a037201_list = f a001223_list where
   f (x:xs@(x':_)) | x == x'   = f xs
                   | otherwise = x : f xs
-- Reinhard Zumkeller, Feb 27 2012

Flatten[Split[Differences[Prime[Range[150]]]]/.{(k_)..}:>k] (* based on a program by Harvey P. Dale, Jun 21 2012 *)

t=0;p=2;forprime(q=3,1e3,if(q-p!=t,print1(q-p", "));t=q-p;p=q) \\ Charles R Greathouse IV, Feb 27 2012

a(n>1) = 2*A373947(n-1). - Gus Wiseman, Sep 16 2024

Offset corrected by Reinhard Zumkeller, Feb 27 2012

A373954 Excess run-compression of standard compositions. Sum of all parts minus sum of compressed parts of the n-th integer composition in standard order.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Jun 27 2024

nonn

The k-th composition in standard order (graded reverse-lexicographic, 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.

We define the (run-) compression of a sequence to be the anti-run obtained by reducing each run of repeated parts to a single part. Alternatively, compression removes all parts equal to the part immediately to their left. For example, (1,1,2,2,1) has compression (1,2,1).

			The excess compression of (2,1,1,3) is 1, so a(92) = 1.

Gus Wiseman, Statistics, classes, and transformations of standard compositions

For length instead of sum we have A124762, counted by A106356.
The opposite for length is A124767, counted by A238279 and A333755.
Positions of zeros are A333489, counted by A003242.
Positions of nonzeros are A348612, counted by A131044.
Compositions counted by this statistic are A373951, opposite A373949.
Compression of standard compositions is A373953.
Positions of ones are A373955.
A037201 gives compression of first differences of primes, halved A373947.
A066099 lists the parts of all compositions in standard order.
A114901 counts compositions with no isolated parts.
A116861 counts partitions by this statistic, by length A116608.
A240085 counts compositions with no unique parts.
A333627 takes the rank of a composition to the rank of its run-lengths.
Cf. A070939, A238130, A238343, A272919, A285981, A333213, A333381, A333382.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Total[stc[n]]-Total[First/@Split[stc[n]]],{n,0,100}]

a(n) = A029837(n) - A373953(n).

A357136 Triangle read by rows where T(n,k) is the number of integer compositions of n with alternating sum k = 0..n. Part of the full triangle A097805.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 3, 0, 3, 0, 1, 0, 6, 0, 4, 0, 1, 10, 0, 10, 0, 5, 0, 1, 0, 20, 0, 15, 0, 6, 0, 1, 35, 0, 35, 0, 21, 0, 7, 0, 1, 0, 70, 0, 56, 0, 28, 0, 8, 0, 1, 126, 0, 126, 0, 84, 0, 36, 0, 9, 0, 1, 0, 252, 0, 210, 0, 120, 0, 45, 0, 10, 0, 1

Offset: 0

Gus Wiseman, Sep 30 2022

A composition of n is a finite sequence of positive integers summing to n.

The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

			Triangle begins:
    1
    0   1
    1   0   1
    0   2   0   1
    3   0   3   0   1
    0   6   0   4   0   1
   10   0  10   0   5   0   1
    0  20   0  15   0   6   0   1
   35   0  35   0  21   0   7   0   1
    0  70   0  56   0  28   0   8   0   1
  126   0 126   0  84   0  36   0   9   0   1
    0 252   0 210   0 120   0  45   0  10   0   1
  462   0 462   0 330   0 165   0  55   0  11   0   1
    0 924   0 792   0 495   0 220   0  66   0  12   0   1
For example, row n = 5 counts the following compositions:
  .  (32)     .  (41)   .  (5)
     (122)       (113)
     (221)       (212)
     (1121)      (311)
     (2111)
     (11111)

The full triangle counting compositions by alternating sum is A097805.
The version for partitions is A103919, full triangle A344651.
This is the right-half of even-indexed rows of A260492.
The triangle without top row and left column is A108044.
Ranking and counting compositions:
- product = sum: A335404, counted by A335405.
- sum = twice alternating sum: A348614, counted by A262977.
- length = alternating sum: A357184, counted by A357182.
- length = absolute value of alternating sum: A357185, counted by A357183.
A003242 counts anti-run compositions, ranked by A333489.
A011782 counts compositions.
A025047 counts alternating compositions, ranked by A345167.
A032020 counts strict compositions, ranked by A233564.
A124754 gives alternating sums of standard compositions.
A238279 counts compositions by sum and number of maximal runs.
Cf. A000120, A051159, A070939, A114220, A114901, A242882, A262046.

Prepend[Table[If[EvenQ[nn],Prepend[#,0],#]&[Riffle[Table[Binomial[nn,k],{k,Floor[nn/2],nn}],0]],{nn,0,10}],{1}]

A344615 Number of compositions of n with no adjacent triples (..., x, y, z, ...) where x <= y <= z.

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 17, 29, 50, 84, 143, 241, 408, 688, 1162, 1959, 3305, 5571, 9393, 15832, 26688, 44980, 75812, 127769, 215338, 362911, 611620, 1030758, 1737131, 2927556, 4933760, 8314754, 14012668, 23615198, 39798098, 67070686, 113032453, 190490542, 321028554

Offset: 0

Gus Wiseman, May 27 2021

nonn

These compositions avoid the weak consecutive pattern (1,2,3), the strict version being A128761.

			The a(1) = 1 through a(6) = 17 compositions:
  (1)  (2)    (3)    (4)      (5)        (6)
       (1,1)  (1,2)  (1,3)    (1,4)      (1,5)
              (2,1)  (2,2)    (2,3)      (2,4)
                     (3,1)    (3,2)      (3,3)
                     (1,2,1)  (4,1)      (4,2)
                     (2,1,1)  (1,3,1)    (5,1)
                              (2,1,2)    (1,3,2)
                              (2,2,1)    (1,4,1)
                              (3,1,1)    (2,1,3)
                              (1,2,1,1)  (2,3,1)
                                         (3,1,2)
                                         (3,2,1)
                                         (4,1,1)
                                         (1,2,1,2)
                                         (1,3,1,1)
                                         (2,1,2,1)
                                         (2,2,1,1)

The case of permutations is A049774.
The strict non-adjacent version is A102726.
The case of permutations of prime indices is A344652.
A001250 counts alternating permutations.
A005649 counts anti-run patterns.
A106356 counts compositions by number of maximal anti-runs.
A114901 counts compositions where each part is adjacent to an equal part.
A344604 counts wiggly compositions with twins.
A344605 counts wiggly patterns with twins.
A344606 counts wiggly permutations of prime factors with twins.
Counting compositions by patterns:
- A003242 avoiding (1,1) adjacent.
- A011782 no conditions.
- A106351 avoiding (1,1) adjacent by sum and length.
- A128695 avoiding (1,1,1) adjacent.
- A128761 avoiding (1,2,3).
- A232432 avoiding (1,1,1).
- A335456 all patterns.
- A335457 all patterns adjacent.
- A335514 matching (1,2,3).
- A344604 weakly avoiding (1,2,3) and (3,2,1) adjacent.
- A344614 avoiding (1,2,3) and (3,2,1) adjacent.
- A344615 weakly avoiding (1,2,3) adjacent.
Cf. A000041, A006330, A008965, A027187, A238279/A333755, A333213, A335464, A335515, A344612, A344619.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MatchQ[#,{_,x_,y_,z_,_}/;x<=y<=z]&]],{n,0,15}]

More terms from Bert Dobbelaere, Jun 12 2021

A373951 Triangle read by rows where T(n,k) is the number of integer compositions of n such that replacing each run of repeated parts with a single part (run-compression) yields a composition of n - k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 3, 0, 1, 0, 4, 2, 1, 1, 0, 7, 4, 4, 0, 1, 0, 14, 5, 6, 5, 1, 1, 0, 23, 14, 10, 10, 6, 0, 1, 0, 39, 26, 29, 12, 14, 6, 1, 1, 0, 71, 46, 54, 40, 19, 16, 9, 0, 1, 0, 124, 92, 96, 82, 64, 22, 22, 8, 1, 1, 0, 214, 176, 204, 144, 137, 82, 30, 26, 10, 0, 1, 0

Offset: 0

Gus Wiseman, Jun 28 2024

			Triangle begins:
    1
    1   0
    1   1   0
    3   0   1   0
    4   2   1   1   0
    7   4   4   0   1   0
   14   5   6   5   1   1   0
   23  14  10  10   6   0   1   0
   39  26  29  12  14   6   1   1   0
   71  46  54  40  19  16   9   0   1   0
  124  92  96  82  64  22  22   8   1   1   0
Row n = 6 counts the following compositions:
  (6)     (411)   (3111)   (33)     (222)  (111111)  .
  (51)    (114)   (1113)   (2211)
  (15)    (1311)  (1221)   (1122)
  (42)    (1131)  (12111)  (21111)
  (24)    (2112)  (11211)  (11112)
  (141)           (11121)
  (321)
  (312)
  (231)
  (213)
  (132)
  (123)
  (2121)
  (1212)
For example, the composition (1,2,2,1) with compression (1,2,1) is counted under T(6,2).

Column k = 0 is A003242 (anti-runs or compressed compositions).
Row-sums are A011782.
Same as A373949 with rows reversed.
Column k = 1 is A373950.
This statistic is represented by A373954, difference A373953.
A114901 counts compositions with no isolated parts.
A116861 counts partitions by compressed sum, by compressed length A116608.
A124767 counts runs in standard compositions, anti-runs A333381.
A240085 counts compositions with no unique parts.
A333755 counts compositions by compressed length.
A373948 represents the run-compression transformation.
Cf. A037201 (halved A373947), A106356, A124762, A238130, A238279, A238343, A285981, A333213, A333382, A333489, A373952.

Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n], Total[First/@Split[#]]==n-k&]],{n,0,10},{k,0,n}]

A005314 For n = 0, 1, 2, a(n) = n; thereafter, a(n) = 2*a(n-1) - a(n-2) + a(n-3).

Original entry on oeis.org

0, 1, 2, 3, 5, 9, 16, 28, 49, 86, 151, 265, 465, 816, 1432, 2513, 4410, 7739, 13581, 23833, 41824, 73396, 128801, 226030, 396655, 696081, 1221537, 2143648, 3761840, 6601569, 11584946, 20330163, 35676949, 62608681, 109870576, 192809420, 338356945, 593775046

Offset: 0

N. J. A. Sloane

Number of compositions of n into parts congruent to {1,2} mod 4. - Vladeta Jovovic, Mar 10 2005
a(n)/a(n-1) tends to A109134; an eigenvalue of the matrix M and a root to the characteristic polynomial. - Gary W. Adamson, May 25 2007
Starting with offset 1 = INVERT transform of (1, 1, 0, 0, 1, 1, 0, 0, ...). - Gary W. Adamson, May 04 2009
a(n-2) is the top left entry of the n-th power of the 3 X 3 matrix [0, 1, 0; 0, 1, 1; 1, 0, 1] or of the 3 X 3 matrix [0, 0, 1; 1, 1, 0; 0, 1, 1]. - R. J. Mathar, Feb 03 2014
Counts closed walks of length (n+2) at a vertex of a unidirectional triangle containing a loop on remaining two vertices. - David Neil McGrath, Sep 15 2014
Also the number of binary words of length n that begin with 1 and avoid the subword 101. a(5) = 9: 10000, 10001, 10010, 10011, 11000, 11001, 11100, 11110, 11111. - Alois P. Heinz, Jul 21 2016
Also the number of binary words of length n-1 such that every two consecutive 0s are immediately followed by at least two consecutive 1s. a(4) = 5: 010, 011, 101, 110, 111. - Jerrold Grossman, May 03 2024

			G.f. = x + 2*x^2 + 3*x^3 + 5*x^4 + 9*x^5 + 16*x^6 + 28*x^7 + 49*x^8 + ...
From _Gus Wiseman_, Nov 25 2019: (Start)
a(n) is the number of subsets of {1..n} containing n such that if x and x + 2 are both in the subset, then so is x + 1. For example, the a(1) = 1 through a(5) = 9 subsets are:
  {1}  {2}    {3}      {4}        {5}
       {1,2}  {2,3}    {1,4}      {1,5}
              {1,2,3}  {3,4}      {2,5}
                       {2,3,4}    {4,5}
                       {1,2,3,4}  {1,2,5}
                                  {1,4,5}
                                  {3,4,5}
                                  {2,3,4,5}
                                  {1,2,3,4,5}
(End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..400
Isha Agarwal, Matvey Borodin, Aidan Duncan, Kaylee Ji, Tanya Khovanova, Shane Lee, Boyan Litchev, Anshul Rastogi, Garima Rastogi, and Andrew Zhao, From Unequal Chance to a Coin Game Dance: Variants of Penney's Game, arXiv:2006.13002 [math.HO], 2020.
Marilena Barnabei, Flavio Bonetti, Niccolò Castronuovo, and Matteo Silimbani, Permutations avoiding a simsun pattern, The Electronic Journal of Combinatorics (2020) Vol. 27, Issue 3, P3.45.
P. Chinn and S. Heubach, Integer Sequences Related to Compositions without 2's, J. Integer Seqs., Vol. 6, 2003.
Hung Viet Chu and Zachary Louis Vasseur, Schreier Sets of Multiples of an Integer, Linear Recurrence, and Pascal Triangle, arXiv:2506.14312 [math.CO], 2025. See Table 1 p. 2.
Christian Ennis, William Holland, Omer Mujawar, Aadit Narayanan, Frank Neubrander, Marie Neubrander, and Christina Simino, Words in Random Binary Sequences I, arXiv:2107.01029 [math.GM], 2021.
R. L. Graham and N. J. A. Sloane, Anti-Hadamard matrices, Linear Alg. Applic., 62 (1984), 113-137.
Jia Huang, A coin flip game and generalizations of Fibonacci numbers, arXiv:2501.07463 [math.CO], 2025. See pp. 8, 10.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 426
L. A. Medina and A. Straub, On multiple and infinite log-concavity, 2013, preprint Annals of Combinatorics, March 2016, Volume 20, Issue 1, pp 125-138.
Denis Neiter and Amsha Proag, Links Between Sums Over Paths in Bernoulli's Triangles and the Fibonacci Numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.8.3.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Bojan Vučković and Miodrag Živković, Row Space Cardinalities Above 2^(n - 2) + 2^(n - 3), ResearchGate, January 2017, p. 3.
Index entries for linear recurrences with constant coefficients, signature (2,-1,1).

Equals row sums of triangle A099557.
Equals row sums of triangle A224838.
Cf. A011973 (starting with offset 1 = Falling diagonal sums of triangle with rows displayed as centered text).
First differences of A005251, shifted twice to the left.
Cf. A003242, A114901, A261041, A274174.

a005314 n = a005314_list !! n
a005314_list = 0 : 1 : 2 : zipWith (+) a005314_list
   (tail $ zipWith (-) (map (2 *) $ tail a005314_list) a005314_list)
-- Reinhard Zumkeller, Oct 14 2011

[0] cat [n le 3 select n else 2*Self(n-1) - Self(n-2) + Self(n-3):n in [1..35]]; // Marius A. Burtea, Oct 24 2019

R:=PowerSeriesRing(Integers(), 36); [0] cat Coefficients(R!( x/(1-2*x+x^2-x^3))); // Marius A. Burtea, Oct 24 2019

A005314 := proc(n)
    option remember ;
    if n <=2 then
        n;
    else
        2*procname(n-1)-procname(n-2)+procname(n-3) ;
    end if;
end proc:
seq(A005314(n),n=0..20) ; # R. J. Mathar, Feb 25 2024

LinearRecurrence[{2, -1, 1}, {0, 1, 2}, 100] (* Vladimir Joseph Stephan Orlovsky, Jul 03 2011 *)
Table[Sum[Binomial[n - Floor[(k + 1)/2], n - Floor[(3 k - 1)/2]], {k, 0, n}], {n, 0, 100}] (* John Molokach, Jul 21 2013 *)
Table[Sum[Binomial[n - Floor[(4 n + 15 - 6 k + (-1)^k)/12], n - Floor[(4 n + 15 - 6 k + (-1)^k)/12] - Floor[(2 n - 1)/3] + k - 1], {k, 1, Floor[(2 n + 2)/3]}], {n, 0, 100}] (* John Molokach, Jul 25 2013 *)
a[ n_] := If[ n < 0, SeriesCoefficient[ x^2 / (1 - x + 2 x^2 - x^3), {x, 0, -n}], SeriesCoefficient[ x / (1 - 2 x + x^2 - x^3), {x, 0, n}]]; (* Michael Somos, Dec 13 2013 *)
RecurrenceTable[{a[0]==0,a[1]==1,a[2]==2,a[n]==2a[n-1]-a[n-2]+a[n-3]},a,{n,40}] (* Harvey P. Dale, May 13 2018 *)
Table[Length[Select[Subsets[Range[n]],MemberQ[#,n]&&!MatchQ[#,{_,x_,y_,_}/;x+2==y]&]],{n,0,10}] (* Gus Wiseman, Nov 25 2019 *)

{a(n) = sum(k=0, (2*n-1)\3, binomial(n-1-k\2, k))}

{a(n) = if( n<0, polcoeff( x^2 / (1 - x + 2*x^2 - x^3) + x * O(x^-n), -n), polcoeff( x / (1 - 2*x + x^2 - x^3) + x * O(x^n), n))}; /* Michael Somos, Sep 18 2012 */

def A005314(n): return sum( binomial(n-k, 2*k+1) for k in range(floor((n+2)/3)) )
[A005314(n) for n in range(51)] # G. C. Greubel, Nov 10 2023

From Paul D. Hanna, Oct 22 2004: (Start)
G.f.: x/(1-2*x+x^2-x^3).
a(n) = Sum_{k=0..[(2n-1)/3]} binomial(n-1-[k/2], k), where [x]=floor(x). (End)
a(n) = Sum_{k=0..n} binomial(n-k, 2*k+1).
23*a_n = 3*P_{2n+2} + 7*P_{2n+1} - 2*P_{2n}, where P_n are the Perrin numbers, A001608. - Don Knuth, Dec 09 2008
G.f. (1-z)*(1+z^2)/(1-2*z+z^2-z^3) for the augmented version 1, 1, 2, 3, 5, 9, 16, 28, 49, 86, 151, ... was given in Simon Plouffe's thesis of 1992.
a(n) = a(n-1) + a(n-2) + a(n-4) = a(n-2) + A049853(n-1) = a(n-1) + A005251(n) = Sum_{i <= n} A005251(i).
a(n) = Sum_{k=0..floor((n-1)/3)} binomial(n-k, 2*k+1). - Richard L. Ollerton, May 12 2004
M^n*[1,0,0] = [a(n-2), a(n-1), a]; where M = the 3 X 3 matrix [0,1,0; 0,0,1; 1,-1,2]. Example M^5*[1,0,0] = [3,5,9]. - Gary W. Adamson, May 25 2007
a(n) = A000931(2*n + 4). - Michael Somos, Sep 18 2012
a(n) = A077954(-n - 2). - Michael Somos, Sep 18 2012
G.f.: 1/( 1 - Sum_{k>=0} x*(x-x^2+x^3)^k ) - 1. - Joerg Arndt, Sep 30 2012
a(n) = Sum_{k=0..n} binomial( n-floor((k+1)/2), n-floor((3k-1)/2) ). - John Molokach, Jul 21 2013
a(n) = Sum_{k=1..floor((2*n+2)/3)} binomial(n - floor((4*n+15-6*k+(-1)^k)/12), n - floor((4*n+15-6*k+(-1)^k)/12) - floor((2*n-1)/3) + k - 1). - John Molokach, Jul 24 2013
a(n) = round(A001608(2n+1)*r) where r is the real root of 23*x^3 - 23*x^2 + 8*x - 1 = 0, r = 0.4114955... - Richard Turk, Oct 24 2019
a(n+2) = n + 2 + Sum_{k=0..n} (n-k)*a(k). - Greg Dresden and Yichen P. Wang, Sep 16 2021
a(n) ~ (19 - r + 11*r^2) / (23 * r^(n-1)), where r = 0.569840290998... is the root of the equation r*(2 - r + r^2) = 1. - Vaclav Kotesovec, Apr 14 2024
a(n) = n*3F2(1/3-n/3,2/3-n/3,1-n/3;-n,3/2;27/4). - R. J. Mathar, Jun 27 2024
If p,q,r are the three solutions to x^3 = 2x^2 - x + 1, then a(n) = p^(n+1)/((p-q)*(p-r)) + q^(n+1)/((q-p)*(q-r)) + r^(n+1)/((r-p)*(r-q)). Compare to similar formula for A005251. - Greg Dresden and AnXing Yang, Aug 19 2025

More terms and additional formulas from Henry Bottomley, Jul 21 2000

Plouffe's g.f. edited by R. J. Mathar, May 12 2008

A348612 Numbers k such that the k-th composition in standard order is not an anti-run, i.e., has adjacent equal parts.

Original entry on oeis.org

3, 7, 10, 11, 14, 15, 19, 21, 23, 26, 27, 28, 29, 30, 31, 35, 36, 39, 42, 43, 46, 47, 51, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 67, 71, 73, 74, 75, 78, 79, 83, 84, 85, 86, 87, 90, 91, 92, 93, 94, 95, 99, 100, 103, 106, 107, 110, 111, 112, 113, 114, 115, 116

Offset: 1

Gus Wiseman, Nov 03 2021

nonn

First differs from A345168 in lacking 37, corresponding to the composition (3,2,1).

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.

			The terms and corresponding standard compositions begin:
     3: (1,1)          35: (4,1,1)        61: (1,1,1,2,1)
     7: (1,1,1)        36: (3,3)          62: (1,1,1,1,2)
    10: (2,2)          39: (3,1,1,1)      63: (1,1,1,1,1,1)
    11: (2,1,1)        42: (2,2,2)        67: (5,1,1)
    14: (1,1,2)        43: (2,2,1,1)      71: (4,1,1,1)
    15: (1,1,1,1)      46: (2,1,1,2)      73: (3,3,1)
    19: (3,1,1)        47: (2,1,1,1,1)    74: (3,2,2)
    21: (2,2,1)        51: (1,3,1,1)      75: (3,2,1,1)
    23: (2,1,1,1)      53: (1,2,2,1)      78: (3,1,1,2)
    26: (1,2,2)        55: (1,2,1,1,1)    79: (3,1,1,1,1)
    27: (1,2,1,1)      56: (1,1,4)        83: (2,3,1,1)
    28: (1,1,3)        57: (1,1,3,1)      84: (2,2,3)
    29: (1,1,2,1)      58: (1,1,2,2)      85: (2,2,2,1)
    30: (1,1,1,2)      59: (1,1,2,1,1)    86: (2,2,1,2)
    31: (1,1,1,1,1)    60: (1,1,1,3)      87: (2,2,1,1,1)

Constant run compositions are counted by A000005, ranked by A272919.
Counting these compositions by sum and length gives A131044.
These compositions are counted by A261983.
The complement is A333489, counted by A003242.
The non-alternating case is A345168, complement A345167.
A011782 counts compositions, strict A032020.
A238279 counts compositions by sum and number of maximal runs.
A274174 counts compositions with equal parts contiguous.
A336107 counts non-anti-run permutations of prime factors.
A345195 counts non-alternating anti-runs, ranked by A345169.
For compositions in standard order (rows of A066099):
- Length is A000120.
- Sum is A070939
- Maximal runs are counted by A124767.
- Strict compositions are ranked by A233564.
- Maximal anti-runs are counted by A333381.
- Runs-resistance is A333628.
Cf. A029931, A048793, A106356, A114901, A167606, A178470, A228351, A244164, A262046, A335452, A335464.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[100],MatchQ[stc[#],{_,x_,x_,_}]&]

A349053 Number of non-weakly alternating integer compositions of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 4, 12, 37, 95, 232, 533, 1198, 2613, 5619, 11915, 25011, 52064, 107694, 221558, 453850, 926309, 1884942, 3825968, 7749312, 15667596, 31628516, 63766109, 128415848, 258365323, 519392582, 1043405306, 2094829709, 4203577778, 8431313237, 16904555958

Offset: 0

Gus Wiseman, Dec 16 2021

nonn

We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either. Then a sequence is (strongly) alternating iff it is a weakly alternating anti-run.

			The a(6) = 12 compositions:
  (1,1,2,2,1)  (1,1,2,3)  (1,2,4)
  (1,2,1,1,2)  (1,2,3,1)  (4,2,1)
  (1,2,2,1,1)  (1,3,2,1)
  (2,1,1,2,1)  (2,1,1,3)
               (3,1,1,2)
               (3,2,1,1)

Andrew Howroyd, Table of n, a(n) for n = 0..1000 (terms 0..55 from Martin Ehrenstein)
Wikipedia, Alternating permutation

Complementary directed versions are A129852/A129853, strong A025048/A025049.
The strong version is A345192.
The complement is counted by A349052.
These compositions are ranked by A349057, strong A345168.
The complementary version for patterns is A349058, strong A345194.
The complementary multiplicative version is A349059, strong A348610.
An unordered version (partitions) is A349061, complement A349060.
The version for ordered prime factorizations is A349797, complement A349056.
The version for patterns is A350138, strong A350252.
The version for ordered factorizations is A350139.
A001250 counts alternating permutations, complement A348615.
A001700 counts compositions of 2n with alternating sum 0.
A003242 counts Carlitz (anti-run) compositions.
A011782 counts compositions, unordered A000041.
A025047 counts alternating compositions, ranked by A345167.
A106356 counts compositions by number of maximal anti-runs.
A344604 counts alternating compositions with twins.
A345164 counts alternating ordered prime factorizations.
A349054 counts strict alternating compositions.
Cf. A102726, A114901, A128761, A261983, A333213, A333755, A344614, A344615, A345165, A345170, A345195, A349799, A349800, A350251, A350252.

wwkQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]<=y[[m+1]],y[[m]]>=y[[m+1]]],{m,1,Length[y]-1}]||And@@Table[If[EvenQ[m],y[[m]]>=y[[m+1]],y[[m]]<=y[[m+1]]],{m,1,Length[y]-1}];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!wwkQ[#]&]],{n,0,10}]

a(n) = A011782(n) - A349052(n).

a(21)-a(35) from Martin Ehrenstein, Jan 08 2022

A349052 Number of weakly alternating compositions of n.

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 28, 52, 91, 161, 280, 491, 850, 1483, 2573, 4469, 7757, 13472, 23378, 40586, 70438, 122267, 212210, 368336, 639296, 1109620, 1925916, 3342755, 5801880, 10070133, 17478330, 30336518, 52653939, 91389518, 158621355, 275313226, 477850887, 829388075

Offset: 0

Gus Wiseman, Nov 29 2021

nonn

We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either. A sequence is alternating iff it is a weakly alternating anti-run.

			The a(5) = 16 compositions:
  (1,1,1,1,1)  (1,1,1,2)  (1,1,3)  (1,4)  (5)
               (1,1,2,1)  (1,2,2)  (2,3)
               (1,2,1,1)  (1,3,1)  (3,2)
               (2,1,1,1)  (2,1,2)  (4,1)
                          (2,2,1)
                          (3,1,1)
The a(6) = 28 compositions:
  (111111)  (11112)  (1113)  (114)  (15)  (6)
            (11121)  (1122)  (132)  (24)
            (11211)  (1131)  (141)  (33)
            (12111)  (1212)  (213)  (42)
            (21111)  (1311)  (222)  (51)
                     (2121)  (231)
                     (2211)  (312)
                     (3111)  (411)

Andrew Howroyd, Table of n, a(n) for n = 0..1000 (Terms 0..55 from Martin Ehrenstein)

The strong case is A025047, ranked by A345167.
The directed versions are A129852 and A129853, strong A025048 and A025049.
The complement is counted by A349053, strong A345192.
The version for permutations of prime indices is A349056, strong A345164.
The complement is ranked by A349057, strong A345168.
The version for patterns is A349058, strong A345194.
The multiplicative version is A349059, strong A348610.
An unordered version (partitions) is A349060, complement A349061.
The non-alternating case is A349800, ranked by A349799.
A001250 counts alternating permutations, complement A348615.
A001700 counts compositions of 2n with alternating sum 0.
A003242 counts Carlitz (anti-run) compositions.
A011782 counts compositions.
A106356 counts compositions by number of maximal anti-runs.
A344604 counts alternating compositions with twins.
A345170 counts partitions w/ an alternating permutation, ranked by A345172.
A349054 counts strict alternating compositions.
Cf. A000041, A008965, A102726, A114901, A128761, A261983, A333213, A333755, A344614, A344615, A345165, A345195.

whkQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]<=y[[m+1]],y[[m]]>=y[[m+1]]],{m,1,Length[y]-1}];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],whkQ[#]||whkQ[-#]&]],{n,0,10}]

C(n,f)={my(M=matrix(n,n,j,k,k>=j), s=M[,n]); for(b=1, n, f=!f; M=matrix(n,n,j,k, if(k1,M[j-k,k-1]) ))); for(k=2, n, M[,k]+=M[,k-1]); s+=M[,n]); s~}
seq(n) = concat([1], C(n,0) + C(n,1) - vector(n,j,numdiv(j))) \\ Andrew Howroyd, Jan 31 2024

a(21)-a(37) from Martin Ehrenstein, Jan 08 2022

A349060 Number of integer partitions of n that are constant or whose part multiplicities, except possibly the first and last, are all even.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 10, 13, 18, 22, 29, 35, 45, 53, 68, 77, 98, 112, 140, 157, 195, 218, 270, 298, 367, 404, 495, 542, 658, 721, 873, 949, 1145, 1245, 1494, 1615, 1934, 2091, 2492, 2688, 3188, 3436, 4068, 4369, 5155, 5537, 6511, 6976, 8186, 8763, 10251, 10962

Offset: 0

Gus Wiseman, Dec 06 2021

nonn

Also the number of weakly alternating integer partitions of n, where we define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either. This sequence looks at the somewhat degenerate case where no strict increases are allowed.

			The a(1) = 1 through a(7) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (32)     (33)      (43)
             (111)  (31)    (41)     (42)      (52)
                    (211)   (221)    (51)      (61)
                    (1111)  (311)    (222)     (322)
                            (2111)   (411)     (331)
                            (11111)  (2211)    (511)
                                     (3111)    (2221)
                                     (21111)   (4111)
                                     (111111)  (22111)
                                               (31111)
                                               (211111)
                                               (1111111)

John Tyler Rascoe, Table of n, a(n) for n = 0..300

Alternating: A025047, ranked by A345167, also A025048 and A025049.
The strong case is A065033, ranked by A167171.
A directed version is A096441.
Non-alternating: A345192, ranked by A345168.
Weakly alternating: A349052, also A129852 and A129853.
Non-weakly alternating: A349053, ranked by A349057.
A version for ordered factorizations is A349059, strong A348610.
The complement is counted by A349061, strong A349801.
These partitions are ranked by the complement of A349794.
The non-strict case is A349795.
A000041 counts integer partitions, ordered A011782.
A001250 counts alternating permutations, complement A348615.
A344604 counts alternating compositions with twins.
A345170 counts partitions w/ an alternating permutation, ranked by A345172.
Cf. A000070, A002865, A003242, A027383, A114901, A117298, A117989, A274230, A345163, A349056, A349796.

Table[Length[Select[IntegerPartitions[n], SameQ@@#||And@@EvenQ/@Take[Length/@Split[#],{2,-2}]&]],{n,0,30}]

A_x(N)={my(x='x+O('x^N), g= 1 + sum(i=1, N, (x^i/(1-x^i)) * (1 + sum(j=i+1, N-i, (x^j/((1-x^j))) / prod(k=1, j-i-1, 1-x^(2*(i+k)))))));
Vec(g)}
A_x(52) \\ John Tyler Rascoe, Mar 20 2024

G.f.: 1 + Sum_{i>0} (x^i/(1-x^i)) * (1 + Sum_{j>i} (x^j/(1-x^j)) / Product_{k=1..j-i-1} (1-x^(2*(i+k)))). - John Tyler Rascoe, Mar 20 2024

cp's OEIS Frontend

A037201 Differences between consecutive primes (A001223) but with repeats omitted.

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A373954 Excess run-compression of standard compositions. Sum of all parts minus sum of compressed parts of the n-th integer composition in standard order.

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A357136 Triangle read by rows where T(n,k) is the number of integer compositions of n with alternating sum k = 0..n. Part of the full triangle A097805.

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A344615 Number of compositions of n with no adjacent triples (..., x, y, z, ...) where x <= y <= z.

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A373951 Triangle read by rows where T(n,k) is the number of integer compositions of n such that replacing each run of repeated parts with a single part (run-compression) yields a composition of n - k.

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A005314 For n = 0, 1, 2, a(n) = n; thereafter, a(n) = 2*a(n-1) - a(n-2) + a(n-3).

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A348612 Numbers k such that the k-th composition in standard order is not an anti-run, i.e., has adjacent equal parts.

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