A333150 -id:A333150 - cp's OEIS frontend

A028859 a(n+2) = 2a(n+1) + 2a(n); a(0) = 1, a(1) = 3.

1, 3, 8, 22, 60, 164, 448, 1224, 3344, 9136, 24960, 68192, 186304, 508992, 1390592, 3799168, 10379520, 28357376, 77473792, 211662336, 578272256, 1579869184, 4316282880, 11792304128, 32217174016, 88018956288, 240472260608, 656982433792, 1794909388800, 4903783645184, 13397386067968

Offset: 0

N. J. A. Sloane

Number of words of length n without adjacent 0's from the alphabet {0,1,2}. For example, a(2) counts 01,02,10,11,12,20,21,22. - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 12 2001
Individually, both this sequence and A002605 are convergents to 1+sqrt(3). Mutually, both sequences are convergents to 2+sqrt(3) and 1+sqrt(3)/2. - Klaus E. Kastberg (kastberg(AT)hotkey.net.au), Nov 04 2001 [Can someone clarify what is meant by the obscure second phrase, "Mutually..."? - M. F. Hasler, Aug 06 2018]
Add a loop at two vertices of the graph C_3=K_3. a(n) counts walks of length n+1 between these vertices. - Paul Barry, Oct 15 2004
Prefaced with a 1 as (1 + x + 3x^2 + 8x^3 + 22x^4 + ...) = 1 / (1 - x - 2x^2 - 3x^3 - 5x^4 - 8x^5 - 13x^6 - 21x^7 - ...). - Gary W. Adamson, Jul 28 2009
Equals row 2 of the array in A180165, and the INVERTi transform of A125145. - Gary W. Adamson, Aug 14 2010
Pisano period lengths: 1, 1, 3, 1, 24, 3, 48, 1, 9, 24, 10, 3, 12, 48, 24, 1, 144, 9, 180, 24, .... - R. J. Mathar, Aug 10 2012
Also the number of independent vertex sets and vertex covers in the n-centipede graph. - Eric W. Weisstein, Sep 21 2017
From Gus Wiseman, May 19 2020: (Start)
Conjecture: Also the number of length n + 1 sequences that cover an initial interval of positive integers and whose non-adjacent parts are weakly decreasing. For example, (3,2,3,1,2) has non-adjacent pairs (3,3), (3,1), (3,2), (2,1), (2,2), (3,2), all of which are weakly decreasing, so is counted under a(11). The a(1) = 1 through a(3) = 8 sequences are:
  (1)  (11)  (111)
       (12)  (121)
       (21)  (211)
             (212)
             (221)
             (231)
             (312)
             (321)
The case of compositions is A333148, or A333150 for strict compositions, or A333193 for strictly decreasing parts. A version for ordered set partitions is A332872. Standard composition numbers of these compositions are A334966. Unimodal normal sequences are A227038. See also: A001045, A001523, A032020, A100471, A100881, A115981, A329398, A332836, A332872.
(End)
Number of 2-compositions of n+1 restricted to parts 1 and 2 (and allowed zeros); see Hopkins & Ouvry reference. - Brian Hopkins, Aug 16 2020
The number of ternary strings of length n not containing 00. Complement of A186244. - R. J. Mathar, Feb 13 2022

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 73).

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Jean-Paul Allouche, Jeffrey Shallit, and Manon Stipulanti, Combinatorics on words and generating Dirichlet series of automatic sequences, arXiv:2401.13524 [math.CO], 2025. See p. 14.
Joerg Arndt, Matters Computational (The Fxtbook), section 14.9 "Strings with no two consecutive zeros", pp.318-320.
C. Bautista-Ramos and C. Guillen-Galvan, Fibonacci numbers of generalized Zykov sums, J. Integer Seq., 15 (2012), #12.7.8.
Moussa Benoumhani, On the Modes of the Independence Polynomial of the Centipede, Journal of Integer Sequences, Vol. 15 (2012), #12.5.1.
D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3 Example 7.
Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
P. Z. Chinn, R. Grimaldi, and S. Heubach, Tiling with Ls and Squares, J. Int. Sequences 10 (2007) #07.2.8.
David Garth and Adam Gouge, Affinely Self-Generating Sets and Morphisms, Journal of Integer Sequences, Article 07.1.5, 10 (2007) 1-13.
Juan B. Gil and Jessica A. Tomasko, Fibonacci colored compositions and applications, arXiv:2108.06462 [math.CO], 2021.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
Brian Hopkins and Stéphane Ouvry, Combinatorics of Multicompositions, arXiv:2008.04937 [math.CO], 2020.
Milan Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
Tanya Khovanova, Recursive Sequences
J. Shallit, Proof of Irvine's conjecture via mechanized guessing, arXiv preprint arXiv:2310.14252 [math.CO], October 22 2023.
Eric Weisstein's World of Mathematics, Centipede Graph
Eric Weisstein's World of Mathematics, Independent Vertex Set
Eric Weisstein's World of Mathematics, Vertex Cover
Index entries for linear recurrences with constant coefficients, signature (2,2).

Cf. A180165, A125145, A026150, A030195, A080040, A083337, A106435, A108898.
Cf. A155020 (same sequence with term 1 prepended).
Cf. A002605.

a028859 n = a028859_list !! n
a028859_list =
   1 : 3 : map (* 2) (zipWith (+) a028859_list (tail a028859_list))
-- Reinhard Zumkeller, Oct 15 2011

a[0]:=1:a[1]:=3:for n from 2 to 24 do a[n]:=2*a[n-1]+2*a[n-2] od: seq(a[n],n=0..24); # Emeric Deutsch

a[n_]:=(MatrixPower[{{1,3},{1,1}},n].{{2},{1}})[[2,1]]; Table[a[n],{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *)
Table[2^(n - 1) Hypergeometric2F1[(1 - n)/2, -n/2, -n, -2], {n, 20}] (* Eric W. Weisstein, Jun 14 2017 *)
LinearRecurrence[{2, 2}, {1, 3}, 20] (* Eric W. Weisstein, Jun 14 2017 *)

a(n)=([1,3;1,1]^n*[2;1])[2,1] \\ Charles R Greathouse IV, Mar 27 2012

A028859(n)=([1,1]*[2,2;1,0]^n)[1] \\ M. F. Hasler, Aug 06 2018

a(n) = a(n-1) + A052945(n) = A002605(n) + A002605(n-1).
G.f.: -(x+1)/(2*x^2+2*x-1).
a(n) = [(1+sqrt(3))^(n+2)-(1-sqrt(3))^(n+2)]/(4*sqrt(3)). - Emeric Deutsch, Feb 01 2005
If p[i]=fibonacci(i+1) and if A is the Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)= det A. - Milan Janjic, May 08 2010
a(n) = 3^n - A186244(n). - Toby Gottfried, Mar 07 2013
E.g.f.: exp(x)*(cosh(sqrt(3)*x) + 2*sinh(sqrt(3)*x)/sqrt(3)). - Stefano Spezia, Mar 02 2024

Definition completed by M. F. Hasler, Aug 06 2018

A218004 Number of equivalence classes of compositions of n where two compositions a,b are considered equivalent if the summands of a can be permuted into the summands of b with an even number of transpositions.

Original entry on oeis.org

1, 1, 2, 4, 6, 9, 14, 19, 27, 37, 51, 67, 91, 118, 156, 202, 262, 334, 430, 543, 690, 867, 1090, 1358, 1696, 2099, 2600, 3201, 3939, 4820, 5899, 7181, 8738, 10590, 12821, 15467, 18644, 22396, 26878, 32166, 38450, 45842, 54599, 64870, 76990, 91181, 107861, 127343, 150182, 176788, 207883

Offset: 0

Geoffrey Critzer, Oct 17 2012

nonn

a(n) = A000041(n) + A000009(n) - 1  where A000041 is the partition numbers and A000009 is the number of partitions into distinct parts.
From Gus Wiseman, Oct 14 2020: (Start)
Also the number of compositions of n that are either strictly increasing or weakly decreasing. For example, the a(1) = 1 through a(6) = 14 compositions are:
  (1)  (2)   (3)    (4)     (5)      (6)
       (11)  (12)   (13)    (14)     (15)
             (21)   (22)    (23)     (24)
             (111)  (31)    (32)     (33)
                    (211)   (41)     (42)
                    (1111)  (221)    (51)
                            (311)    (123)
                            (2111)   (222)
                            (11111)  (321)
                                     (411)
                                     (2211)
                                     (3111)
                                     (21111)
                                     (111111)
A007997 counts only compositions of length 3.
A329398 appears to be the weakly increasing version.
A333147 is the strictly decreasing version.
A333255 union A114994 ranks these compositions using standard compositions (A066099).
A337482 counts the complement.
(End)

			a(4) = 6 because the 6 classes can be represented by: 4, 3+1, 1+3, 2+2, 2+1+1, 1+1+1+1.

A000009 counts strictly increasing compositions, ranked by A333255.
A000041 counts weakly decreasing compositions, ranked by A114994.
A001523 counts unimodal compositions (strict: A072706).
A007318 and A097805 count compositions by length.
A032020 counts strict compositions, ranked by A233564.
A332834 counts compositions not increasing nor decreasing (strict: A333149).
Cf. A115981, A225620, A332578, A332833, A332874, A333150, A333190, A333191, A333256, A337483, A337484.

nn=50;p=CoefficientList[Series[Product[1/(1-x^i),{i,1,nn}],{x,0,nn}],x];d= CoefficientList[Series[Sum[Product[x^i/(1-x^i),{i,1,k}],{k,0,nn}],{x,0,nn}],x];p+d-1
(* second program *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Less@@#||GreaterEqual@@#&]],{n,0,15}] (* Gus Wiseman, Oct 14 2020 *)

A333147 Number of compositions of n that are either strictly increasing or strictly decreasing.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 7, 9, 11, 15, 19, 23, 29, 35, 43, 53, 63, 75, 91, 107, 127, 151, 177, 207, 243, 283, 329, 383, 443, 511, 591, 679, 779, 895, 1023, 1169, 1335, 1519, 1727, 1963, 2225, 2519, 2851, 3219, 3631, 4095, 4607, 5179, 5819, 6527, 7315, 8193, 9163

Offset: 0

Gus Wiseman, May 16 2020

nonn

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

			The a(1) = 1 through a(9) = 15 compositions:
  (1)  (2)  (3)    (4)    (5)    (6)      (7)      (8)      (9)
            (1,2)  (1,3)  (1,4)  (1,5)    (1,6)    (1,7)    (1,8)
            (2,1)  (3,1)  (2,3)  (2,4)    (2,5)    (2,6)    (2,7)
                          (3,2)  (4,2)    (3,4)    (3,5)    (3,6)
                          (4,1)  (5,1)    (4,3)    (5,3)    (4,5)
                                 (1,2,3)  (5,2)    (6,2)    (5,4)
                                 (3,2,1)  (6,1)    (7,1)    (6,3)
                                          (1,2,4)  (1,2,5)  (7,2)
                                          (4,2,1)  (1,3,4)  (8,1)
                                                   (4,3,1)  (1,2,6)
                                                   (5,2,1)  (1,3,5)
                                                            (2,3,4)
                                                            (4,3,2)
                                                            (5,3,1)
                                                            (6,2,1)

Eric Weisstein's World of Mathematics, Unimodal Sequence

Strict partitions are A000009.
Unimodal compositions are A001523 (strict: A072706).
Strict compositions are A032020.
The non-strict version appears to be A329398.
Partitions with incr. or decr. run-lengths are A332745 (strict: A333190).
Compositions with incr. or decr. run-lengths are A332835 (strict: A333191).
The complement is counted by A333149 (non-strict: A332834).
Cf. A059204, A072705, A072707, A115981, A332285, A332578, A332746, A332831, A332833, A332874, A333150.

Mathematica
```
Table[2*PartitionsQ[n]-1,{n,0,30}]
```

a(n) = 2*A000009(n) - 1.

A333149 Number of strict compositions of n that are neither increasing nor decreasing.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 4, 4, 8, 12, 38, 42, 72, 98, 150, 298, 372, 542, 760, 1070, 1428, 2600, 3120, 4550, 6050, 8478, 10976, 15220, 23872, 29950, 41276, 55062, 74096, 97148, 129786, 167256, 256070, 314454, 429338, 556364, 749266, 955746, 1275016, 1618054

Offset: 0

Gus Wiseman, May 16 2020

nonn

A composition of n is a finite sequence of positive integers summing to n. It is strict if there are no repeated parts.

			The a(6) = 4 through a(9) = 12 compositions:
  (1,3,2)  (1,4,2)  (1,4,3)  (1,5,3)
  (2,1,3)  (2,1,4)  (1,5,2)  (1,6,2)
  (2,3,1)  (2,4,1)  (2,1,5)  (2,1,6)
  (3,1,2)  (4,1,2)  (2,5,1)  (2,4,3)
                    (3,1,4)  (2,6,1)
                    (3,4,1)  (3,1,5)
                    (4,1,3)  (3,2,4)
                    (5,1,2)  (3,4,2)
                             (3,5,1)
                             (4,2,3)
                             (5,1,3)
                             (6,1,2)

Eric Weisstein's World of Mathematics, Unimodal Sequence

The non-strict case is A332834.
The complement is counted by A333147.
Strict partitions are A000009.
Strict compositions are A032020.
Non-unimodal strict compositions are A072707.
Strict partitions with increasing or decreasing run-lengths are A333190.
Strict compositions with increasing or decreasing run-lengths are A333191.
Unimodal compositions are A001523, with strict case A072706.
Cf. A059204, A115981, A227038, A329398, A332745, A332746, A332831, A332833, A332835, A332874, A333150, A333192.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@#&&!Greater@@#&&!Less@@#&]],{n,0,10}]

a(n) = A032020(n) - 2*A000009(n) + 1.

A333148 Number of compositions of n whose non-adjacent parts are weakly decreasing.

Original entry on oeis.org

1, 1, 2, 4, 7, 12, 19, 30, 46, 69, 102, 149, 214, 304, 428, 596, 823, 1127, 1532, 2068, 2774, 3697, 4900, 6460, 8474, 11061, 14375, 18600, 23970, 30770, 39354, 50153, 63702, 80646, 101783, 128076, 160701, 201076, 250933, 312346, 387832, 480409, 593716, 732105, 900810, 1106063, 1355336, 1657517, 2023207, 2464987, 2997834, 3639464

Offset: 0

Gus Wiseman, May 16 2020

nonn

			The a(1) = 1 through a(6) = 19 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)
       (11)  (12)   (13)    (14)     (15)
             (21)   (22)    (23)     (24)
             (111)  (31)    (32)     (33)
                    (121)   (41)     (42)
                    (211)   (131)    (51)
                    (1111)  (212)    (141)
                            (221)    (222)
                            (311)    (231)
                            (1211)   (312)
                            (2111)   (321)
                            (11111)  (411)
                                     (1311)
                                     (2121)
                                     (2211)
                                     (3111)
                                     (12111)
                                     (21111)
                                     (111111)
For example, (2,3,1,2) is such a composition, because the non-adjacent pairs of parts are (2,1), (2,2), (3,2), all of which are weakly decreasing.

Alois P. Heinz, Table of n, a(n) for n = 0..700

Unimodal compositions are A001523.
The case of normal sequences appears to be A028859.
A version for ordered set partitions is A332872.
The case of strict compositions is A333150.
The version for strictly decreasing parts is A333193.
Standard composition numbers (A066099) of these compositions are A334966.
Cf. A056242, A059204, A072706, A107429, A115981, A329398, A332578, A332669, A332673, A332724, A332834.

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

def a333148(n): return number_of_partitions(n) + sum( Partitions(m, max_part=l, length=k).cardinality() * Partitions(n-m-l^2, min_length=k+2*l).cardinality() for l in range(1, (n+1).isqrt()) for m in range((n-l^2-2*l)*l//(l+1)+1) for k in range(ceil(m/l), min(m,n-m-l^2-2*l)+1) ) # Max Alekseyev, Oct 31 2024

See Sage code for the formula. - Max Alekseyev, Oct 31 2024

Edited and terms a(21)-a(51) added by Max Alekseyev, Oct 30 2024

A333193 Number of compositions of n whose non-adjacent parts are strictly decreasing.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 21, 29, 40, 53, 71, 93, 122, 158, 204, 260, 332, 419, 528, 661, 825, 1023, 1267, 1560, 1916, 2344, 2860, 3476, 4217, 5097, 6147, 7393, 8872, 10618, 12685, 15115, 17977, 21336, 25276, 29882, 35271, 41551, 48872, 57385, 67277, 78745, 92040

Offset: 0

Gus Wiseman, May 18 2020

nonn

			The a(1) = 1 through a(7) = 15 compositions:
  (1)  (2)   (3)   (4)    (5)    (6)     (7)
       (11)  (12)  (13)   (14)   (15)    (16)
             (21)  (22)   (23)   (24)    (25)
                   (31)   (32)   (33)    (34)
                   (211)  (41)   (42)    (43)
                          (221)  (51)    (52)
                          (311)  (231)   (61)
                                 (312)   (241)
                                 (321)   (322)
                                 (411)   (331)
                                 (2211)  (412)
                                         (421)
                                         (511)
                                         (2311)
                                         (3211)
For example, (2,3,1,2) is not such a composition, because the non-adjacent pairs of parts are (2,1), (2,2), (3,2), not all of which are strictly decreasing, while (2,4,1,1) is such a composition, because the non-adjacent pairs of parts are (2,1), (2,1), (4,1), all of which are strictly decreasing.

Andrew Howroyd, Table of n, a(n) for n = 0..1000

A version for ordered set partitions is A332872.
The case of strict compositions is A333150.
The case of normal sequences appears to be A001045.
Unimodal compositions are A001523, with strict case A072706.
Strict compositions are A032020.
Partitions with strictly increasing run-lengths are A100471.
Partitions with strictly decreasing run-lengths are A100881.
Compositions with weakly decreasing non-adjacent parts are A333148.
Compositions with strictly increasing run-lengths are A333192.
Cf. A059204, A072707, A115981, A227038, A332834, A332836, A333191, A334966.

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

\\ p is all, q is those ending in an unreversed singleton.
seq(n)={my(q=O(x*x^n), p=1+q); for(k=1, n, [p,q] = [p*(1 + x^k + x^(2*k)) + q*x^k, q + p*x^k] ); Vec(p)} \\ Andrew Howroyd, Apr 17 2021

Terms a(21) and beyond from Andrew Howroyd, Apr 17 2021

A334966 Numbers k such that the k-th composition in standard order (row k of A066099) has weakly decreasing non-adjacent parts.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 27, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 47, 48, 49, 51, 55, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 85, 86, 87

Offset: 1

Gus Wiseman, May 18 2020

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.

The complement starts: 14, 26, 28, 29, 30, 44, 46, 50, ...

			The sequence together with the corresponding compositions begins:
   0: ()           17: (4,1)          37: (3,2,1)
   1: (1)          18: (3,2)          38: (3,1,2)
   2: (2)          19: (3,1,1)        39: (3,1,1,1)
   3: (1,1)        20: (2,3)          40: (2,4)
   4: (3)          21: (2,2,1)        41: (2,3,1)
   5: (2,1)        22: (2,1,2)        42: (2,2,2)
   6: (1,2)        23: (2,1,1,1)      43: (2,2,1,1)
   7: (1,1,1)      24: (1,4)          45: (2,1,2,1)
   8: (4)          25: (1,3,1)        47: (2,1,1,1,1)
   9: (3,1)        27: (1,2,1,1)      48: (1,5)
  10: (2,2)        31: (1,1,1,1,1)    49: (1,4,1)
  11: (2,1,1)      32: (6)            51: (1,3,1,1)
  12: (1,3)        33: (5,1)          55: (1,2,1,1,1)
  13: (1,2,1)      34: (4,2)          63: (1,1,1,1,1,1)
  15: (1,1,1,1)    35: (4,1,1)        64: (7)
  16: (5)          36: (3,3)          65: (6,1)
For example, (2,3,1,2) is such a composition because the non-adjacent pairs are (2,1), (2,2), (3,2), all of which are weakly decreasing, so 166 is in the sequence

The case of normal sequences appears to be A028859.
Strict compositions are A032020.
A version for ordered set partitions is A332872.
These compositions are enumerated by A333148.
The strict case is enumerated by A333150.
Cf. A072706, A072707, A227038, A332834, A333193.

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

cp's OEIS Frontend

A028859 a(n+2) = 2*a(n+1) + 2*a(n); a(0) = 1, a(1) = 3.

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A218004 Number of equivalence classes of compositions of n where two compositions a,b are considered equivalent if the summands of a can be permuted into the summands of b with an even number of transpositions.

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A333147 Number of compositions of n that are either strictly increasing or strictly decreasing.

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A333149 Number of strict compositions of n that are neither increasing nor decreasing.

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A333148 Number of compositions of n whose non-adjacent parts are weakly decreasing.

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A333193 Number of compositions of n whose non-adjacent parts are strictly decreasing.

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A334966 Numbers k such that the k-th composition in standard order (row k of A066099) has weakly decreasing non-adjacent parts.

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Programs

Mathematica

A028859 a(n+2) = 2a(n+1) + 2a(n); a(0) = 1, a(1) = 3.