A003242 -id:A003242 - cp's OEIS frontend

A124762 Number of levels for compositions in standard order.

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

Offset: 0

Franklin T. Adams-Watters, Nov 06 2006

The standard order of compositions is given by A066099.

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. This sequence gives the number of adjacent equal terms in the n-th composition in standard order. Alternatively, a(n) is one fewer than the number of maximal anti-runs in the same composition, where anti-runs are sequences without any adjacent equal terms. For example, the 1234567th composition in standard order is (3,2,1,2,2,1,2,5,1,1,1) with anti-runs ((3,2,1,2),(2,1,2,5,1),(1),(1)), so a(1234567) = 4 - 1 = 3. - Gus Wiseman, Apr 08 2020

			Composition number 11 is 2,1,1; 2>1=1, so a(11) = 1.
The table starts:
  0
  0
  0 1
  0 0 0 2
  0 0 1 1 0 0 1 3
  0 0 0 1 0 1 0 2 0 0 1 1 1 1 2 4
  0 0 0 1 1 0 0 2 0 0 2 2 0 0 1 3 0 0 0 1 0 1 0 2 1 1 2 2 2 2 3 5

Cf. A066099, A124760, A124761, A124763, A124764, A011782 (row lengths), A059570 (row sums).
Anti-runs summing to n are counted by A003242(n).
A triangle counting maximal anti-runs of compositions is A106356.
A triangle counting maximal runs of compositions is A238279.
Partitions whose first differences are an anti-run are A238424.
All of the following pertain to compositions in standard order (A066099):
- Weakly decreasing runs are counted by A124765.
- Weakly increasing runs are counted by A124766.
- Equal runs are counted by A124767.
- Strictly increasing runs are counted by A124768.
- Strictly decreasing runs are counted by A124769.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Normal compositions are A333217.
- Adjacent unequal pairs are counted by A333382.
- Anti-runs are A333489.
Cf. A000120, A029931, A048793, A059893, A070939, A114994, A225620, A228351.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length[Select[Partition[stc[n],2,1],SameQ@@#&]],{n,0,100}] (* Gus Wiseman, Apr 08 2020 *)

For a composition b(1),...,b(k), a(n) = Sum_{1<=i=1

For n > 0, a(n) = A333381(n) - 1. - Gus Wiseman, Apr 08 2020

A329738 Number of compositions of n whose run-lengths are all equal.

Original entry on oeis.org

1, 1, 2, 4, 6, 8, 19, 24, 45, 75, 133, 215, 401, 662, 1177, 2035, 3587, 6190, 10933, 18979, 33339, 58157, 101958, 178046, 312088, 545478, 955321, 1670994, 2925717, 5118560, 8960946, 15680074, 27447350, 48033502, 84076143, 147142496, 257546243, 450748484, 788937192

Offset: 0

Gus Wiseman, Nov 20 2019

nonn

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

			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)
                    (1111)  (131)    (51)
                            (212)    (123)
                            (11111)  (132)
                                     (141)
                                     (213)
                                     (222)
                                     (231)
                                     (312)
                                     (321)
                                     (1122)
                                     (1212)
                                     (2121)
                                     (2211)
                                     (111111)

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

Compositions with relatively prime run-lengths are A000740.
Compositions with equal multiplicities are A098504.
Compositions with equal differences are A175342.
Compositions with distinct run-lengths are A329739.
Cf. A003242, A008965, A107429, A164707, A238130, A242882, A274174, A329745, A329766.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],SameQ@@Length/@Split[#]&]],{n,0,10}]

seq(n)={my(b=Vec(1/(1 - sum(k=1, n, x^k/(1+x^k) + O(x*x^n)))-1)); concat([1], vector(n, k, sumdiv(k, d, b[d])))} \\ Andrew Howroyd, Dec 30 2020

a(n) = Sum_{d|n} A003242(d).

a(n) = A329745(n) + A000005(n).

A335452 Number of separations (Carlitz compositions or anti-runs) of the prime indices of n.

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 2, 2, 1, 0, 0, 2, 0, 1, 1, 6, 1, 0, 2, 2, 2, 2, 1, 2, 2, 0, 1, 6, 1, 1, 1, 2, 1, 0, 0, 1, 2, 1, 1, 0, 2, 0, 2, 2, 1, 6, 1, 2, 1, 0, 2, 6, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 2, 6, 1, 0, 0, 2, 1, 6, 2, 2, 2

Offset: 1

Gus Wiseman, Jun 21 2020

nonn

The first term that is not a factorial number is a(180) = 12.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
A separation (or Carlitz composition) of a multiset is a permutation with no adjacent equal parts.
a(n) depends only on the prime signature of n. - Andrew Howroyd, Feb 03 2021

			The a(n) separations for n = 2, 6, 30, 180:
  (1)  (12)  (123)  (12123)
       (21)  (132)  (12132)
             (213)  (12312)
             (231)  (12321)
             (312)  (13212)
             (321)  (21213)
                    (21231)
                    (21312)
                    (21321)
                    (23121)
                    (31212)
                    (32121)

Andrew Howroyd, Table of n, a(n) for n = 1..4096
Wikipedia, Permutation pattern

Separations are counted by A003242 and ranked by A333489.
Patterns are counted by A000670 and ranked by A333217.
Permutations of prime indices are counted by A008480.
Inseparable partitions are counted by A325535 and ranked by A335448.
Cf. A000961, A005117, A056239, A112798, A181796, A261962, A333221, A335451, A335454, A335465, A335489.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Permutations[primeMS[n]],!MatchQ[#,{_,x_,x_,_}]&]],{n,100}]

F(i, j, r, t) = {sum(k=max(0, i-j), min(min(i,t), (i-j+t)\2), binomial(i, k)*binomial(r-i+1, t+i-j-2*k)*binomial(t-1, k-i+j))}
count(sig)={my(s=vecsum(sig), r=0, v=[1]); for(p=1, #sig, my(t=sig[p]); v=vector(s-r-t+1, j, sum(i=1, #v, v[i]*F(i-1, j-1, r, t))); r += t); v[1]}
a(n)={count(factor(n)[,2])} \\ Andrew Howroyd, Feb 03 2021

A345167 Numbers k such that the k-th composition in standard order is alternating.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 8, 9, 12, 13, 16, 17, 18, 20, 22, 24, 25, 32, 33, 34, 38, 40, 41, 44, 45, 48, 49, 50, 54, 64, 65, 66, 68, 70, 72, 76, 77, 80, 81, 82, 88, 89, 96, 97, 98, 102, 108, 109, 128, 129, 130, 132, 134, 140, 141, 144, 145, 148, 152, 153, 160, 161, 162

Offset: 1

Gus Wiseman, Jun 15 2021

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.

A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no alternating permutations, even though it does have the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2).

			The terms together with their binary indices begin:
      1: (1)         25: (1,3,1)       66: (5,2)
      2: (2)         32: (6)           68: (4,3)
      4: (3)         33: (5,1)         70: (4,1,2)
      5: (2,1)       34: (4,2)         72: (3,4)
      6: (1,2)       38: (3,1,2)       76: (3,1,3)
      8: (4)         40: (2,4)         77: (3,1,2,1)
      9: (3,1)       41: (2,3,1)       80: (2,5)
     12: (1,3)       44: (2,1,3)       81: (2,4,1)
     13: (1,2,1)     45: (2,1,2,1)     82: (2,3,2)
     16: (5)         48: (1,5)         88: (2,1,4)
     17: (4,1)       49: (1,4,1)       89: (2,1,3,1)
     18: (3,2)       50: (1,3,2)       96: (1,6)
     20: (2,3)       54: (1,2,1,2)     97: (1,5,1)
     22: (2,1,2)     64: (7)           98: (1,4,2)
     24: (1,4)       65: (6,1)        102: (1,3,1,2)

Wikipedia, Alternating permutation

These compositions are counted by A025047, complement A345192.
The complement is A345168.
Partitions with a permutation of this type: A345170, complement A345165.
Factorizations with a permutation of this type: A348379.
A001250 counts alternating permutations, complement A348615.
A003242 counts anti-run compositions.
A345164 counts alternating permutations of prime indices.
A345194 counts alternating patterns, with twins A344605.
Statistics of standard compositions:
- Length is A000120.
- Constant runs are A124767.
- Heinz number is A333219.
- Number of maximal anti-runs is A333381.
- Runs-resistance is A333628.
- Number of distinct parts is A334028.
Classes of standard compositions:
- Weakly decreasing compositions (partitions) are A114994.
- Weakly increasing compositions (multisets) are A225620.
- Anti-runs are A333489.
- Non-alternating anti-runs are A345169.
Cf. A025048, A025049, A059893, A106356, A238279, A335448, A344604, A344615, A344653, A344742, A345163, A348377.

stc[n_]:=Differences[Prepend[Join@@Position[ Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
wigQ[y_]:=Or[Length[y]==0,Length[Split[y]] ==Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Select[Range[0,100],wigQ@*stc]

A374249 Numbers k such that the k-th composition in standard order has its equal parts contiguous.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 26, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 47, 48, 50, 52, 56, 58, 60, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 78, 79, 80, 81, 83, 84, 85

Offset: 1

Gus Wiseman, Jul 13 2024

nonn

These are compositions avoiding the patterns (1,2,1) and (2,1,2).

			The terms together with their standard compositions begin:
   0: ()
   1: (1)
   2: (2)
   3: (1,1)
   4: (3)
   5: (2,1)
   6: (1,2)
   7: (1,1,1)
   8: (4)
   9: (3,1)
  10: (2,2)
  11: (2,1,1)
  12: (1,3)
  14: (1,1,2)
  15: (1,1,1,1)
  16: (5)
See A374253 for the complement: 13, 22, 25, 27, 29, ...

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

The strict (also anti-run) case is A233564, counted by A032020.
Compositions of this type are counted by A274174.
Permutations of prime indices of this type are counted by A333175.
The complement is A374253 (anti-run A374254), counted by A335548.
A003242 counts anti-run compositions, ranks A333489.
A011782 counts compositions.
A066099 lists compositions in standard order.
A124767 counts runs in standard compositions, anti-runs A333381.
A333755 counts compositions by number of runs.
A335454 counts patterns matched by standard compositions.
A335462 counts (1,2,1)- and (2,1,2)-matching permutations of prime indices.
- A335470 counts (1,2,1)-matching compositions, ranks A335466.
- A335471 counts (1,2,1)-avoiding compositions, ranks A335467.
- A335472 counts (2,1,2)-matching compositions, ranks A335468.
- A335473 counts (2,1,2)-avoiding compositions, ranks A335469.
Cf. A106356, A124762, A238130, A238279, A261982, A272919, A333382, A335450, A335460, A335524, A335525.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],UnsameQ@@First/@Split[stc[#]]&]

Equals A335467 /\ A335469.

A333382 Number of adjacent unequal parts in the n-th composition in standard-order.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Mar 24 2020

nonn

For n > 0, a(n) is one fewer than the number of maximal runs of the n-th composition in standard-order.

			The 46th composition in standard order is (2,1,1,2), with maximal runs ((2),(1,1),(2)), so a(46) = 3 - 1 = 2.

Wikipedia, Longest increasing subsequence

Indices of first appearances (not counting 0) are A113835.
Partitions whose 0-appended first differences are a run are A007862.
Partitions whose first differences are a run are A049988.
A triangle counting maximal anti-runs of compositions is A106356.
A triangle counting maximal runs of compositions is A238279.
All of the following pertain to compositions in standard order (A066099):
- Adjacent equal pairs are counted by A124762.
- Weakly decreasing runs are counted by A124765.
- Weakly increasing runs are counted by A124766.
- Equal runs are counted by A124767.
- Strictly increasing runs are counted by A124768.
- Strictly decreasing runs are counted by A124769.
- Strict compositions are ranked by A233564.
- Constant compositions are ranked by A272919.
- Normal compositions are ranked by A333217.
- Anti-runs are ranked by A333489.
- Anti-runs are counted by A333381.
Cf. A000005, A000120, A003242, A029931, A048793, A059893, A070939, A114994, A225620, A228351, A238424.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length[Select[Partition[stc[n],2,1],UnsameQ@@#&]],{n,0,100}]

For n > 0, a(n) = A124767(n) - 1.

A006498 a(n) = a(n-1) + a(n-3) + a(n-4), a(0) = a(1) = a(2) = 1, a(3) = 2.

Original entry on oeis.org

1, 1, 1, 2, 4, 6, 9, 15, 25, 40, 64, 104, 169, 273, 441, 714, 1156, 1870, 3025, 4895, 7921, 12816, 20736, 33552, 54289, 87841, 142129, 229970, 372100, 602070, 974169, 1576239, 2550409, 4126648, 6677056, 10803704, 17480761, 28284465, 45765225, 74049690, 119814916

Offset: 0

N. J. A. Sloane

Number of compositions of n into 1's, 3's and 4's. - Len Smiley, May 08 2001
The sum of any two alternating terms (terms separated by one term) produces a number from the Fibonacci sequence. (e.g. 4+9=13, 9+25=34, 6+15=21, etc.) Taking square roots starting from the first term and every other term after, we get the Fibonacci sequence. - Sreyas Srinivasan (sreyas_srinivasan(AT)hotmail.com), May 02 2002
(1 + x + 2*x^2 + x^3)/(1 - x - x^3 - x^4) = 1 + 2*x + 4*x^2 + 6*x^3 + 9*x^4 + 15*x^5 + 25*x^6 + 40*x^7 + ... is the g.f. for the number of binary strings of length where neither 101 nor 111 occur. [Lozansky and Rousseau] Or, equivalently, where neither 000 nor 010 occur.
Equivalently, a(n+2) is the number of length-n binary strings with no two set bits with distance 2; see fxtbook link. - Joerg Arndt, Jul 10 2011
a(n) is the number of words written with the letters "a" and "b", with the following restriction: any "a" must be followed by at least two letters, the second of which is a "b". - Bruno Petazzoni (bpetazzoni(AT)ac-creteil.fr), Oct 31 2005. [This is also equivalent to the previous two conditions.]
Let a(0) = 1, then a(n) = partial products of Product_{n>2} (F(n)/F(n-1))^2 = 1*1*2*2*(3/2)*(3/2)*(5/3)*(5/3)*(8/5)*(8/5)*.... E.g., a(7) = 15 = 1*1*1*2*2*(3/2)*(3/2)*(5/3). - Gary W. Adamson, Dec 13 2009
Number of permutations satisfying -k <= p(i) - i <= r and p(i)-i not in I, i=1..n, with k=1, r=3, I={1}. - Vladimir Baltic, Mar 07 2012
The 2-dimensional version, which counts sets of pairs no two of which are separated by graph-distance 2, is A273461. - Gus Wiseman, Nov 27 2019
a(n+1) is the number of multus bitstrings of length n with no runs of 4 ones. - Steven Finch, Mar 25 2020

			G.f. = 1 + x + x^2 + 2*x^3 + 4*x^4 + 6*x^5 + 9*x^6 + 15*x^7 + 25*x^8 + 40*x^9 + ...
From _Gus Wiseman_, Nov 27 2019: (Start)
The a(2) = 1 through a(7) = 15 subsets with no two elements differing by 2:
  {}  {}   {}     {}     {}     {}
      {1}  {1}    {1}    {1}    {1}
           {2}    {2}    {2}    {2}
           {1,2}  {3}    {3}    {3}
                  {1,2}  {4}    {4}
                  {2,3}  {1,2}  {5}
                         {1,4}  {1,2}
                         {2,3}  {1,4}
                         {3,4}  {1,5}
                                {2,3}
                                {2,5}
                                {3,4}
                                {4,5}
                                {1,2,5}
                                {1,4,5}
(End)

E. Lozansky and C. Rousseau, Winning Solutions, Springer, 1996; see pp. 157 and 172.
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..500
Katharine A. Ahrens, Combinatorial Applications of the k-Fibonacci Numbers: A Cryptographically Motivated Analysis, Ph. D. thesis, North Carolina State University (2020).
Michael A. Allen, Connections between Combinations Without Specified Separations and Strongly Restricted Permutations, Compositions, and Bit Strings, arXiv:2409.00624 [math.CO], 2024. See pp. 16, 18.
D. Applegate, M. LeBrun, and N. J. A. Sloane, Dismal Arithmetic, J. Int. Seq. 14 (2011) # 11.9.8.
Said Amrouche and Hacène Belbachir, Unimodality and linear recurrences associated with rays in the Delannoy triangle, Turkish Journal of Mathematics (2019) Vol. 44, 118-130.
Joerg Arndt, Matters Computational (The Fxtbook), section 14.10.1, p.320.
Vladimir Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics Vol. 4, No 1 (2010), 119-135.
V. Baltic, Applications of the finite state automata for counting restricted permutations and variations, Yug. J. Oper. Res. 22 (2012) 183-198, Sec. 3
G. E. Bergum and V. E. Hoggatt, Jr., A combinatorial problem involving recursive sequences and tridiagonal matrices, Fib. Quart., 16 (1978), 113-118.
M. El-Mikkawy, T. Sogabe, A new family of k-Fibonacci numbers, Appl. Math. Comput. 215 (2010) 4456-4461.
K. Edwards, A Pascal-like triangle related to the tribonacci numbers, Fib. Q., 46/47 (2008/2009), 18-25.
Steven Finch, Cantor-solus and Cantor-multus distributions, arXiv:2003.09458 [math.CO], 2020.
T. Guardia and D. Jiménez, Fiboquadratic Sequences and Extensions of the Cassini Identity Raised From the Study of Rithmomachia, arXiv preprint arXiv:1509.03177 [math.HO], 2015-2016.
John Konvalina, Yi-Hsin Liu, subsets without q-separation and binomial products of Fibonacci numbers, J. Comb. Theo. A 57 (2) (1991) 306-310, T_n.
Andreas M. Hinz and Paul K. Stockmeyer, Precious Metal Sequences and Sierpinski-Type Graphs, J. Integer Seq., Vol 25 (2022), Article 22.4.8.
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
M. Tetiva, Subsets that make no difference d, Mathematics Magazine 84 (2011), no. 4, 300-301.
Dominika Závacká, Cristina Dalfó, and Miquel Angel Fiol, Integer sequences from k-iterated line digraphs, CEUR: Proc. 24th Conf. Info. Tech. - Appl. and Theory (ITAT 2024) Vol 3792, 156-161. See p. 161, Table 2.
Index entries for sequences related to Chebyshev polynomials.
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (1,0,1,1).

Cf. A060945 (for 1's, 2's and 4's). Essentially the same as A074677.
Diagonal sums of number triangle A059259.
Cf. A000032, A000034, A000045, A001654, A007598, A209398.
Numbers whose binary expansion has no subsequence (1,0,1) are A048716.
Cf. A003242, A005251, A027383, A167606, A261041, A273461, A329860, A329871.

a006498 n = a006498_list !! n
a006498_list = 1 : 1 : 1 : 2 : zipWith (+) (drop 3 a006498_list)
   (zipWith (+) (tail a006498_list) a006498_list)
-- Reinhard Zumkeller, Apr 07 2012

[ n eq 1 select 1 else n eq 2 select 1 else n eq 3 select 1 else n eq 4 select 2 else Self(n-1)+Self(n-3)+ Self(n-4): n in [1..40] ]; // Vincenzo Librandi, Aug 20 2011

LinearRecurrence[{1,0,1,1},{1,1,1,2},50] (* Harvey P. Dale, Jul 13 2011 *)
Table[Fibonacci[Floor[n/2] + 2]^Mod[n, 2]*Fibonacci[Floor[n/2] + 1]^(2 - Mod[n, 2]), {n, 0, 40}] (* David Nacin, Feb 29 2012 *)
a[ n_] := Fibonacci[ Quotient[ n+2, 2]] Fibonacci[ Quotient[ n+3, 2]] (* Michael Somos, Jan 19 2014 *)
Table[Length[Select[Subsets[Range[n]],!MatchQ[#,{_,x_,_,y_,_}/;x+2==y]&]],{n,10}] (* Gus Wiseman, Nov 27 2019 *)

{a(n) = fibonacci( (n+2)\2 ) * fibonacci( (n+3)\2 )} /* Michael Somos, Mar 10 2004 */

PARI
```
Vec(1/(1-x-x^3-x^4)+O(x^66))
```

def a(n, adict={0:1, 1:1, 2:1, 3:2}):
    if n in adict:
        return adict[n]
    adict[n]=a(n-1)+a(n-3)+a(n-4)
    return adict[n] # David Nacin, Mar 07 2012

G.f.: 1 / ((1 + x^2) * (1 - x - x^2)); a(2*n) = F(n+1)^2, a(2*n - 1) = F(n+1)*F(n). a(n) = a(-4-n) * (-1)^n. - Michael Somos, Mar 10 2004
The g.f. -(1+z+2*z^2+z^3)/((z^2+z-1)*(1+z^2)) for the truncated version 1, 2, 4, 6, 9, 15, 25, 40, ... was given in the Simon Plouffe thesis of 1992. [edited by R. J. Mathar, May 13 2008]
From Vladeta Jovovic, May 03 2002: (Start)
a(n) = round((-(1/5)*sqrt(5) - 1/5)*(-2*1/(-sqrt(5)+1))^n/(-sqrt(5)+1) + ((1/5)*sqrt(5) - 1/5)*(-2*1/( sqrt(5)+1))^n/(sqrt(5)+1)).
G.f.: 1/(1-x-x^2)/(1+x^2). (End)
a(n) = (-i)^n*Sum{k=0..n} U(n-2k, i/2) where i^2=-1. - Paul Barry, Nov 15 2003
a(n) = Sum_{k=0..floor(n/2)} (-1)^k*F(n-2k+1). - Paul Barry, Oct 12 2007
F(floor(n/2) + 2)^(n mod 2)*F(floor(n/2) + 1)^(2 - (n mod 2)) where F(n) is the n-th Fibonacci number. - David Nacin, Feb 29 2012
a(2*n - 1) = A001654(n), a(2*n) = A007598(n+1). - Michael Somos, Mar 10 2004
a(n+1)*a(n+3) = a(n)*a(n+2) + a(n+1)*a(n+2) for all n in Z. - Michael Somos, Jan 19 2014
a(n) = round(1/(1/F(n+2) + 2/F(n+3))), where F(n) = A000045, and 0.5 is rounded to 1. - Richard R. Forberg, Aug 04 2014
5*a(n) = (-1)^floor(n/2)*A000034(n+1) + A000032(n+2). - R. J. Mathar, Sep 16 2017
a(n) = Sum_{j=0..floor(n/3)} Sum_{k=0..j} binomial(n-3j,k)*binomial(j,k)*2^k. - Tony Foster III, Sep 18 2017
E.g.f.: (2*cos(x) + sin(x) + exp(x/2)*(3*cosh(sqrt(5)*x/2) + sqrt(5)*sinh(sqrt(5)*x/2)))/5. - Stefano Spezia, Mar 12 2024

A001522 Number of n-stacks with strictly receding walls, or the number of Type A partitions of n in the sense of Auluck (1951).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 5, 7, 10, 14, 19, 26, 35, 47, 62, 82, 107, 139, 179, 230, 293, 372, 470, 591, 740, 924, 1148, 1422, 1756, 2161, 2651, 3244, 3957, 4815, 5844, 7075, 8545, 10299, 12383, 14859, 17794, 21267, 25368, 30207, 35902, 42600, 50462, 59678, 70465, 83079, 97800, 114967, 134956, 158205, 185209, 216546, 252859

Offset: 0

N. J. A. Sloane

Also number of partitions of n with positive crank (n>=2), cf. A064391. - Vladeta Jovovic, Sep 30 2001
Number of smooth weakly unimodal compositions of n into positive parts such that the first and last part are 1 (smooth means that successive parts differ by at most one), see example. Dropping the requirement for unimodality gives A186085. - Joerg Arndt, Dec 09 2012
Number of weakly unimodal compositions of n where the maximal part m appears at least m times, see example. - Joerg Arndt, Jun 11 2013
Also weakly unimodal compositions of n with first part 1, maximal up-step 1, and no consecutive up-steps; see example. The smooth weakly unimodal compositions are recovered by shifting all rows above the bottom row to the left by one position with respect to the next lower row. - Joerg Arndt, Mar 30 2014
It would seem from Stanley that he regards a(0)=0 for this sequence and A001523. - Michael Somos, Feb 22 2015
From Gus Wiseman, Mar 30 2021: (Start)
Also the number of odd-length compositions of n with alternating parts strictly decreasing. These are finite odd-length sequences q of positive integers summing to n such that q(i) > q(i+2) for all possible i. The even-length version is A064428. For example, the a(1) = 1 through a(9) = 14 compositions are:
  (1)  (2)  (3)  (4)    (5)    (6)    (7)    (8)    (9)
                 (211)  (221)  (231)  (241)  (251)  (261)
                        (311)  (312)  (322)  (332)  (342)
                               (321)  (331)  (341)  (351)
                               (411)  (412)  (413)  (423)
                                      (421)  (422)  (432)
                                      (511)  (431)  (441)
                                             (512)  (513)
                                             (521)  (522)
                                             (611)  (531)
                                                    (612)
                                                    (621)
                                                    (711)
                                                    (32211)
(End)
In the Ferrers diagram of a partition x of n, count the dots in each diagonal parallel to the main diagonal (starting at the top-right, say). The result diag(x) is a smooth weakly unimodal composition of n into positive parts such that the first and last part are 1. For example, diag(5541) = 11233221. The function diag is many-to-one; the size of its codomain as a set is a(n). If diag(x) = diag(y), each hook of x can be slid by the same amount past the main diagonal to get y. For example, diag(5541) = diag(44331). - George Beck, Sep 26 2021
From Gus Wiseman, May 23 2022: (Start)
Conjecture: Also the number of integer partitions y of n with a fixed point y(i) = i. These partitions are ranked by A352827. The conjecture is stated at A238395, but Resta tells me he may not have had a proof. The a(1) = 1 through a(8) = 10 partitions are:
  (1)  (11)  (111)  (22)    (32)     (42)      (52)       (62)
                    (1111)  (221)    (222)     (322)      (422)
                            (11111)  (321)     (421)      (521)
                                     (2211)    (2221)     (2222)
                                     (111111)  (3211)     (3221)
                                               (22111)    (4211)
                                               (1111111)  (22211)
                                                          (32111)
                                                          (221111)
                                                          (11111111)
Note that these are not the same partitions (compare A352827 to A352874), only the same count (apparently).
(End)
The above conjecture is true. See Section 4 of the Blecher-Knopfmacher paper in the Links section. - Jeremy Lovejoy, Sep 26 2022

			For a(6)=5 we have the following stacks:
.x... ..x.. ...x. .xx.
xxxxx xxxxx xxxxx xxxx xxxxxx
.
From _Joerg Arndt_, Dec 09 2012: (Start)
There are a(9) = 14 smooth weakly unimodal compositions of 9:
01:   [ 1 1 1 1 1 1 1 1 1 ]
02:   [ 1 1 1 1 1 1 2 1 ]
03:   [ 1 1 1 1 1 2 1 1 ]
04:   [ 1 1 1 1 2 1 1 1 ]
05:   [ 1 1 1 1 2 2 1 ]
06:   [ 1 1 1 2 1 1 1 1 ]
07:   [ 1 1 1 2 2 1 1 ]
08:   [ 1 1 2 1 1 1 1 1 ]
09:   [ 1 1 2 2 1 1 1 ]
10:   [ 1 1 2 2 2 1 ]
11:   [ 1 2 1 1 1 1 1 1 ]
12:   [ 1 2 2 1 1 1 1 ]
13:   [ 1 2 2 2 1 1 ]
14:   [ 1 2 3 2 1 ]
(End)
From _Joerg Arndt_, Jun 11 2013: (Start)
There are a(9) = 14 weakly unimodal compositions of 9 where the maximal part m appears at least m times:
01:  [ 1 1 1 1 1 1 1 1 1 ]
02:  [ 1 1 1 1 1 2 2 ]
03:  [ 1 1 1 1 2 2 1 ]
04:  [ 1 1 1 2 2 1 1 ]
05:  [ 1 1 1 2 2 2 ]
06:  [ 1 1 2 2 1 1 1 ]
07:  [ 1 1 2 2 2 1 ]
08:  [ 1 2 2 1 1 1 1 ]
09:  [ 1 2 2 2 1 1 ]
10:  [ 1 2 2 2 2 ]
11:  [ 2 2 1 1 1 1 1 ]
12:  [ 2 2 2 1 1 1 ]
13:  [ 2 2 2 2 1 ]
14:  [ 3 3 3 ]
(End)
From _Joerg Arndt_, Mar 30 2014: (Start)
There are a(9) = 14 compositions of 9 with first part 1, maximal up-step 1, and no consecutive up-steps:
01:  [ 1 1 1 1 1 1 1 1 1 ]
02:  [ 1 1 1 1 1 1 1 2 ]
03:  [ 1 1 1 1 1 1 2 1 ]
04:  [ 1 1 1 1 1 2 1 1 ]
05:  [ 1 1 1 1 1 2 2 ]
06:  [ 1 1 1 1 2 1 1 1 ]
07:  [ 1 1 1 1 2 2 1 ]
08:  [ 1 1 1 2 1 1 1 1 ]
09:  [ 1 1 1 2 2 1 1 ]
10:  [ 1 1 1 2 2 2 ]
11:  [ 1 1 2 1 1 1 1 1 ]
12:  [ 1 1 2 2 1 1 1 ]
13:  [ 1 1 2 2 2 1 ]
14:  [ 1 1 2 2 3 ]
(End)
G.f. = 1 + x + x^2 + x^3 + 2*x^4 + 3*x^5 + 5*x^6 + 7*x^7 + 10*x^8 + 14*x^9 + ...

G. E. Andrews, The reasonable and unreasonable effectiveness of number theory in statistical mechanics, pp. 21-34 of S. A. Burr, ed., The Unreasonable Effectiveness of Number Theory, Proc. Sympos. Appl. Math., 46 (1992). Amer. Math. Soc.
G. E. Andrews, Three-quadrant Ferrers graphs, Indian J. Math., 42 (No. 1, 2000), 1-7.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1999; see section 2.5 on page 76.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
F. C. Auluck, On some new types of partitions associated with generalized Ferrers graphs, Proc. Cambridge Philos. Soc. 47, (1951), 679-686.
F. C. Auluck, On some new types of partitions associated with generalized Ferrers graphs (annotated scanned copy)
A. Blecher and A. Knopfmacher, Fixed points and matching points in partitions, Ramanujan J. 58 (2022), 23-41.
Sergi Elizalde, Symmetric peaks and symmetric valleys in Dyck paths, arXiv:2008.05669 [math.CO], 2020.
Erich Friedman, Illustration of initial terms
A. D. Sokal, The leading root of the partial theta function, arXiv preprint arXiv:1106.1003 [math.CO], 2011.
E. M. Wright, Stacks, III, Quart. J. Math. Oxford, 23 (1972), 153-158.

Cf. A000041, A059776, A001523, A001524.
A version for permutations is A002467, complement A000166.
The case of zero crank is A064410, ranked by A342192.
The case of nonnegative crank is A064428, ranked by A352873.
A strict version is A352829, complement A352828.
Conjectured to be column k = 1 of A352833.
These partitions (positive crank) are ranked by A352874.
A000700 counts self-conjugate partitions, ranked by A088902.
A064391 counts partitions by crank.
A115720 and A115994 count partitions by their Durfee square.
A257989 gives the crank of the partition with Heinz number n.
Counting compositions: A003242, A114921, A238351, A342527, A342528, A342532.
Fixed points of reversed partitions: A238352, A238394, A238395, A352822, A352830, A352872.
Cf. A008292, A114088, A118199, A325187, A352827, A352832.

b:= proc(n, i, t) option remember; `if`(n<=0, `if`(i=1, 1, 0),
      `if`(n<0 or i<1, 0, b(n-i, i, t)+b(n-(i-1), i-1, false)+
      `if`(t, b(n-(i+1), i+1, t), 0)))
    end:
a:= n-> b(n-1, 1, true):
seq(a(n), n=0..70);  # Alois P. Heinz, Feb 26 2014
# second Maple program:
A001522 := proc(n)
    local r,a;
    a := 0 ;
    if n = 0 then
        return 1 ;
    end if;
    for r from 1 do
        if r*(r+1) > 2*n then
            return a;
        else
            a := a-(-1)^r*combinat[numbpart](n-r*(r+1)/2) ;
        end if;
    end do:
end proc: # R. J. Mathar, Mar 07 2015

max = 50; f[x_] := 1 + Sum[-(-1)^k*x^(k*(k+1)/2), {k, 1, max}] / Product[(1-x^k), {k, 1, max}]; CoefficientList[ Series[ f[x], {x, 0, max}], x] (* Jean-François Alcover, Dec 27 2011, after g.f. *)
b[n_, i_, t_] := b[n, i, t] = If[n <= 0, If[i == 1, 1, 0], If[n<0 || i<1, 0, b[n-i, i, t] + b[n - (i-1), i-1, False] + If[t, b[n - (i+1), i+1, t], 0]]]; a[n_] := b[n-1, 1, True]; Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Dec 01 2015, after Alois P. Heinz *)
Flatten[{1, Table[Sum[(-1)^(j-1)*PartitionsP[n-j*((j+1)/2)], {j, 1, Floor[(Sqrt[8*n + 1] - 1)/2]}], {n, 1, 60}]}] (* Vaclav Kotesovec, Sep 26 2016 *)
ici[q_]:=And@@Table[q[[i]]>q[[i+2]],{i,Length[q]-2}];
Table[If[n==0,1,Length[Select[Join@@Permutations/@Select[IntegerPartitions[n],OddQ@*Length],ici]]],{n,0,15}] (* Gus Wiseman, Mar 30 2021 *)

{a(n) = if( n<1, n==0, polcoeff( sum(k=1, (sqrt(1+8*n) - 1)\2, -(-1)^k * x^((k + k^2)/2)) / eta(x + x * O(x^n)), n))}; /* Michael Somos, Jul 22 2003 */

N=66; q='q+O('q^N);
Vec( 1 + sum(n=1, N, q^(n^2)/(prod(k=1,n-1,1-q^k)^2*(1-q^n)) ) ) \\ Joerg Arndt, Dec 09 2012

def A001522(n):
    if n < 4: return 1
    return (number_of_partitions(n) - [p.crank() for p in Partitions(n)].count(0))/2
[A001522(n) for n in range(30)]  # Peter Luschny, Sep 15 2014

a(n) = (A000041(n) - A064410(n)) / 2 for n>=2.
G.f.: 1 + ( Sum_{k>=1} -(-1)^k * x^(k*(k+1)/2) ) / ( Product_{k>=1} 1-x^k ).
G.f.: 1 + ( Sum_{n>=1} q^(n^2) / ( ( Product_{k=1..n-1} 1-q^k )^2 * (1-q^n) ) ). - Joerg Arndt, Dec 09 2012
a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*sqrt(3)*n) [Auluck, 1951]. - Vaclav Kotesovec, Sep 26 2016
a(n) = A000041(n) - A064428(n). - Gus Wiseman, Mar 30 2021
a(n) = A064428(n) - A064410(n). - Gus Wiseman, May 23 2022

a(0) changed from 0 to 1 by Joerg Arndt, Mar 30 2014

Edited definition. - N. J. A. Sloane, Mar 31 2021

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cp's OEIS Frontend

A329739 Number of compositions of n whose run-lengths are all different.

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A373949 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 k.

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A124762 Number of levels for compositions in standard order.

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A329738 Number of compositions of n whose run-lengths are all equal.

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A335452 Number of separations (Carlitz compositions or anti-runs) of the prime indices of n.

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A345167 Numbers k such that the k-th composition in standard order is alternating.

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A374249 Numbers k such that the k-th composition in standard order has its equal parts contiguous.

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A333382 Number of adjacent unequal parts in the n-th composition in standard-order.

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