A317086 -id:A317086 - cp's OEIS frontend

A227038 Number of (weakly) unimodal compositions of n where all parts 1, 2, ..., m appear where m is the largest part.

1, 1, 1, 3, 4, 7, 13, 19, 30, 44, 71, 98, 147, 205, 294, 412, 575, 783, 1077, 1456, 1957, 2634, 3492, 4627, 6082, 7980, 10374, 13498, 17430, 22451, 28767, 36806, 46803, 59467, 75172, 94839, 119285, 149599, 187031, 233355, 290340, 360327, 446222, 551251, 679524, 835964, 1026210

Offset: 0

Joerg Arndt, Jun 28 2013

nonn

			There are a(8) = 30 such compositions of 8:
01:  [ 1 1 1 1 1 1 1 1 ]
02:  [ 1 1 1 1 1 1 2 ]
03:  [ 1 1 1 1 1 2 1 ]
04:  [ 1 1 1 1 2 1 1 ]
05:  [ 1 1 1 1 2 2 ]
06:  [ 1 1 1 2 1 1 1 ]
07:  [ 1 1 1 2 2 1 ]
08:  [ 1 1 1 2 3 ]
09:  [ 1 1 1 3 2 ]
10:  [ 1 1 2 1 1 1 1 ]
11:  [ 1 1 2 2 1 1 ]
12:  [ 1 1 2 2 2 ]
13:  [ 1 1 2 3 1 ]
14:  [ 1 1 3 2 1 ]
15:  [ 1 2 1 1 1 1 1 ]
16:  [ 1 2 2 1 1 1 ]
17:  [ 1 2 2 2 1 ]
18:  [ 1 2 2 3 ]
19:  [ 1 2 3 1 1 ]
20:  [ 1 2 3 2 ]
21:  [ 1 3 2 1 1 ]
22:  [ 1 3 2 2 ]
23:  [ 2 1 1 1 1 1 1 ]
24:  [ 2 2 1 1 1 1 ]
25:  [ 2 2 2 1 1 ]
26:  [ 2 2 3 1 ]
27:  [ 2 3 1 1 1 ]
28:  [ 2 3 2 1 ]
29:  [ 3 2 1 1 1 ]
30:  [ 3 2 2 1 ]
From _Gus Wiseman_, Mar 05 2020: (Start)
The a(1) = 1 through a(6) = 13 compositions:
  (1)  (11)  (12)   (112)   (122)    (123)
             (21)   (121)   (221)    (132)
             (111)  (211)   (1112)   (231)
                    (1111)  (1121)   (321)
                            (1211)   (1122)
                            (2111)   (1221)
                            (11111)  (2211)
                                     (11112)
                                     (11121)
                                     (11211)
                                     (12111)
                                     (21111)
                                     (111111)
(End)

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 0..10000
Wikipedia, Composition (combinatorics)
Eric Weisstein's World of Mathematics, Unimodal Sequence
Wikipedia, Unimodality

Cf. A001523 (unimodal compositions), A001522 (smooth unimodal compositions with first and last part 1), A001524 (unimodal compositions such that each up-step is by at most 1 and first part is 1).
Organizing by length rather than sum gives A007052.
The complement is counted by A332743.
The case of run-lengths of partitions is A332577, with complement A332579.
Compositions covering an initial interval are A107429.
Non-unimodal compositions are A115981.
Cf. A000009, A055932, A072704, A317086, A329766, A332578, A332669, A332670.

b:= proc(n,i) option remember;
      `if`(i>n, 0, `if`(irem(n, i)=0, 1, 0)+
      add(b(n-i*j, i+1)*(j+1), j=1..n/i))
    end:
a:= n-> `if`(n=0, 1, b(n, 1)):
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 26 2014

b[n_, i_] := b[n, i] = If[i>n, 0, If[Mod[n, i] == 0, 1, 0] + Sum[b[n-i*j, i+1]*(j+1), {j, 1, n/i}]]; a[n_] := If[n==0, 1, b[n, 1]]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Apr 09 2015, after Alois P. Heinz *)
normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],normQ[#]&&unimodQ[#]&]],{n,0,10}] (* Gus Wiseman, Mar 05 2020 *)

a(n) ~ c * exp(Pi*sqrt(r*n)) / n, where r = 0.9409240878664458093345791978063..., c = 0.05518035191234679423222212249... - Vaclav Kotesovec, Mar 04 2020

a(n) + A332743(n) = 2^(n - 1). - Gus Wiseman, Mar 05 2020

A332280 Number of integer partitions of n with unimodal run-lengths.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 41, 55, 75, 97, 129, 166, 215, 273, 352, 439, 557, 692, 865, 1066, 1325, 1614, 1986, 2413, 2940, 3546, 4302, 5152, 6207, 7409, 8862, 10523, 12545, 14814, 17562, 20690, 24397, 28615, 33645, 39297, 46009, 53609, 62504, 72581, 84412

Offset: 0

Gus Wiseman, Feb 18 2020

nonn

First differs from A000041 at a(10) = 41, A000041(10) = 42.

A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing followed by a weakly decreasing sequence.

			The a(10) = 41 partitions (A = 10) are:
  (A)     (61111)   (4321)     (3211111)
  (91)    (55)      (43111)    (31111111)
  (82)    (541)     (4222)     (22222)
  (811)   (532)     (42211)    (222211)
  (73)    (5311)    (421111)   (2221111)
  (721)   (5221)    (4111111)  (22111111)
  (7111)  (52111)   (3331)     (211111111)
  (64)    (511111)  (3322)     (1111111111)
  (631)   (442)     (331111)
  (622)   (4411)    (32221)
  (6211)  (433)     (322111)
Missing from this list is only (33211).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Unimodal Sequence

The complement is counted by A332281.
Heinz numbers of these partitions are the complement of A332282.
Taking 0-appended first-differences instead of run-lengths gives A332283.
The normal case is A332577.
The opposite version is A332638.
Unimodal compositions are A001523.
Unimodal normal sequences are A007052.
Numbers whose unsorted prime signature is unimodal are A332288.
Cf. A007052, A025065, A072706, A100883, A115981, A227038, A317086, A328509, A329398, A332284, A332285, A332294, A332578, A332579.

b:= proc(n, i, m, t) option remember; `if`(n=0, 1,
     `if`(i<1, 0, add(b(n-i*j, i-1, j, t and j>=m),
      j=1..min(`if`(t, [][], m), n/i))+b(n, i-1, m, t)))
    end:
a:= n-> b(n$2, 0, true):
seq(a(n), n=0..65);  # Alois P. Heinz, Feb 20 2020

unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]]
Table[Length[Select[IntegerPartitions[n],unimodQ[Length/@Split[#]]&]],{n,0,30}]
(* Second program: *)
b[n_, i_, m_, t_] := b[n, i, m, t] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1, j, t && j >= m], {j, 1, Min[If[t, Infinity, m], n/i]}] + b[n, i - 1, m, t]]];
a[n_] := b[n, n, 0, True];
a /@ Range[0, 65] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

A317245 Number of supernormal integer partitions of n.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 4, 4, 1, 3, 3, 4, 2, 4, 5, 6, 6, 10, 7, 10, 9, 9, 10, 11, 12, 12, 21, 12, 18, 17, 21, 19, 28, 23, 28, 26, 27, 24, 32, 29, 36, 34, 46, 42, 55, 48, 65, 65, 74, 70, 88, 81, 83, 103, 112, 129, 153, 157, 190, 205, 210, 242, 283, 276, 321

Offset: 0

Gus Wiseman, Jul 24 2018

nonn

An integer partition is supernormal if either (1) it is of the form 1^n for some n >= 0, or (2a) it spans an initial interval of positive integers, and (2b) its multiplicities, sorted in weakly decreasing order, are themselves a supernormal integer partition.

			The a(10) = 4 supernormal integer partitions are (4321), (33211), (322111), (1111111111).
The a(21) = 10 supernormal integer partitions:
  (654321),
  (4443321),
  (44432211), (44333211), (44332221),
  (4432221111), (4333221111), (4332222111),
  (433322211),
  (111111111111111111111).

Cf. A000041, A181819, A182850, A182857, A275870, A304660, A305563, A317081, A317086, A317088, A317246.

supnrm[q_]:=Or[q=={}||Union[q]=={1},And[Union[q]==Range[Max[q]],supnrm[Sort[Length/@Split[q],Greater]]]];
Table[Length[Select[IntegerPartitions[n],supnrm]],{n,0,30}]

A317491 Number of fully normal integer partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 6, 10, 12, 17, 21, 30, 33, 46, 50, 68, 77, 100, 112, 146, 167, 201, 234, 290, 326, 400, 456, 545, 622, 744, 845, 1004, 1153, 1351, 1551, 1819, 2103, 2434, 2808, 3248, 3735, 4304, 4943, 5661, 6506, 7446, 8499, 9657, 11070, 12505, 14325, 16183

Offset: 0

Gus Wiseman, Jul 30 2018

nonn

An integer partition is fully normal if either it is of the form (1,1,...,1) or its multiplicities span an initial interval of positive integers and, sorted in weakly decreasing order, are themselves fully normal.

			The a(6) = 6 fully normal partitions are (6), (51), (42), (411), (321), (111111). Missing from this list are (33), (3111), (222), (2211), (21111).

Cf. A181819, A182850, A182857, A275870, A304660, A305563, A317081, A317086, A317088, A317246, A317492, A317493.

fulnrmQ[ptn_]:=With[{qtn=Sort[Length/@Split[ptn],Greater]},Or[ptn=={}||Union[ptn]=={1},And[Union[qtn]==Range[Max[qtn]],fulnrmQ[qtn]]]];
Table[Length[Select[IntegerPartitions[n],fulnrmQ]],{n,0,30}]

a(n) = A317245(n) iff n is 1 or a prime number.

A329395 Numbers whose binary expansion without the most significant (first) digit has Lyndon and co-Lyndon factorizations of equal lengths.

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 10, 13, 15, 16, 22, 25, 31, 32, 36, 42, 46, 49, 53, 59, 63, 64, 76, 82, 94, 97, 109, 115, 127, 128, 136, 148, 156, 162, 166, 169, 170, 172, 181, 182, 190, 193, 201, 202, 211, 213, 214, 217, 221, 227, 235, 247, 255, 256, 280, 292, 306, 308

Offset: 1

Gus Wiseman, Nov 13 2019

nonn

We define the Lyndon product of two or more finite sequences to be the lexicographically maximal sequence obtainable by shuffling the sequences together. For example, the Lyndon product of (231) with (213) is (232131), the product of (221) with (213) is (222131), and the product of (122) with (2121) is (2122121). A Lyndon word is a finite sequence that is prime with respect to the Lyndon product. Equivalently, a Lyndon word is a finite sequence that is lexicographically strictly less than all of its cyclic rotations. Every finite sequence has a unique (orderless) factorization into Lyndon words, and if these factors are arranged in lexicographically decreasing order, their concatenation is equal to their Lyndon product. For example, (1001) has sorted Lyndon factorization (001)(1).
Similarly, the co-Lyndon product is the lexicographically minimal sequence obtainable by shuffling the sequences together, and a co-Lyndon word is a finite sequence that is prime with respect to the co-Lyndon product, or, equivalently, a finite sequence that is lexicographically strictly greater than all of its cyclic rotations. For example, (1001) has sorted co-Lyndon factorization (1)(100).
Conjecture: also numbers k such that the k-th composition in standard order (A066099) is a palindrome, cf. A025065, A242414, A317085, A317086, A317087, A335373. - Gus Wiseman, Jun 06 2020

			The sequence of terms together with their trimmed binary expansions and their co-Lyndon and Lyndon factorizations begins:
   1:      () =               0 = 0
   2:     (0) =             (0) = (0)
   3:     (1) =             (1) = (1)
   4:    (00) =          (0)(0) = (0)(0)
   7:    (11) =          (1)(1) = (1)(1)
   8:   (000) =       (0)(0)(0) = (0)(0)(0)
  10:   (010) =         (0)(10) = (01)(0)
  13:   (101) =         (10)(1) = (1)(01)
  15:   (111) =       (1)(1)(1) = (1)(1)(1)
  16:  (0000) =    (0)(0)(0)(0) = (0)(0)(0)(0)
  22:  (0110) =        (0)(110) = (011)(0)
  25:  (1001) =        (100)(1) = (1)(001)
  31:  (1111) =    (1)(1)(1)(1) = (1)(1)(1)(1)
  32: (00000) = (0)(0)(0)(0)(0) = (0)(0)(0)(0)(0)
  36: (00100) =     (0)(0)(100) = (001)(0)(0)
  42: (01010) =     (0)(10)(10) = (01)(01)(0)
  46: (01110) =       (0)(1110) = (0111)(0)
  49: (10001) =       (1000)(1) = (1)(0001)
  53: (10101) =     (10)(10)(1) = (1)(01)(01)
  59: (11011) =     (110)(1)(1) = (1)(1)(011)
  63: (11111) = (1)(1)(1)(1)(1) = (1)(1)(1)(1)(1)

Lyndon and co-Lyndon compositions are (both) counted by A059966.
Numbers whose reversed binary expansion is Lyndon are A328596.
Numbers whose binary expansion is co-Lyndon are A275692.
Cf. A001037, A060223, A102659, A211100, A329131, A329312, A329313, A329318, A329326, A329394, A329398.
Cf. A329358, A329399, A329400, A329401.

lynQ[q_]:=Array[Union[{q, RotateRight[q, #]}]=={q, RotateRight[q, #]}&, Length[q]-1, 1, And];
lynfac[q_]:=If[Length[q]==0, {}, Function[i, Prepend[lynfac[Drop[q, i]], Take[q, i]]][Last[Select[Range[Length[q]], lynQ[Take[q, #]]&]]]];
colynQ[q_]:=Array[Union[{RotateRight[q, #], q}]=={RotateRight[q, #], q}&, Length[q]-1, 1, And];
colynfac[q_]:=If[Length[q]==0, {}, Function[i, Prepend[colynfac[Drop[q, i]], Take[q, i]]]@Last[Select[Range[Length[q]], colynQ[Take[q, #]]&]]];
Select[Range[100],Length[lynfac[Rest[IntegerDigits[#,2]]]]==Length[colynfac[Rest[IntegerDigits[#,2]]]]&]

A317087 Numbers whose prime factors span an initial interval of prime numbers and whose sequence of prime multiplicities is a palindrome.

Original entry on oeis.org

1, 2, 4, 6, 8, 16, 30, 32, 36, 64, 90, 128, 210, 216, 256, 270, 300, 512, 810, 900, 1024, 1296, 2048, 2310, 2430, 2700, 2940, 3000, 3150, 4096, 7290, 7776, 8100, 8192, 9000, 11550, 16384, 21870, 24300, 27000, 30000, 30030, 32768, 41160, 44100, 46656, 47250, 48510

Offset: 1

Gus Wiseman, Jul 21 2018

nonn

3^m*10^k for k, m > 0 are terms of this sequence. - Chai Wah Wu, Jun 23 2020

			The sequence of rows of A296150 indexed by the terms of this sequence begins: (1), (11), (21), (111), (1111), (321), (11111), (2211), (111111), (3221), (1111111), (4321), (222111), (11111111), (32221), (33211), (111111111), (322221), (332211).

Giovanni Resta, Table of n, a(n) for n = 1..10000
Wikipedia, Palindrome

Cf. A025065, A055932, A124010, A133808, A242414, A296150, A317085, A317086.

nrmpalQ[n_]:=With[{f=If[n==1,{},FactorInteger[n]]}, And[PrimePi/@ Sort[First/@f] == Range[ Length[f]], Reverse[Last/@f] == Last/@f]]; Select[Range[100],nrmpalQ]
upto = 10^20; pL[n_] := Block[{p = Prime@Range@n, h = Ceiling[n/2]}, Take[p, h] Reverse@ If[n == 2 h, Take[p, -h], Prepend[ Take[p, 1-h], 1]]]; ric[v_, p_] := If[p == {}, AppendTo[L, v], Block[{w = v}, While[w <= upto, ric[w, Rest@ p]; w *= First@ p]]]; np = 1; L = {1}; While[(b = Times @@ Prime[Range@ np]) <= upto, ric[b, pL[np++]]]; Sort[L] (* Giovanni Resta, Jun 23 2020 *)

from sympy import factorint, primepi
A317087_list = [1]
for n in range(1,10**5):
    d = factorint(n)
    k, l = sorted(d.keys()), len(d)
    if l > 0 and l == primepi(max(d)):
        for i in range(l//2):
            if d[k[i]] != d[k[l-i-1]]:
                break
        else:
            A317087_list.append(n) # Chai Wah Wu, Jun 23 2020

A317256 Number of alternately co-strong integer partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 11, 13, 19, 25, 35, 42, 61, 74, 98, 122, 161, 194, 254, 304, 388, 472, 589, 700, 878, 1044, 1278, 1525, 1851, 2182, 2651, 3113, 3735, 4389, 5231, 6106, 7278, 8464, 9995, 11631, 13680, 15831, 18602, 21463, 25068, 28927, 33654, 38671, 44942, 51514

Offset: 0

Gus Wiseman, Jul 25 2018

nonn

A sequence is alternately co-strong if either it is empty, equal to (1), or its run-lengths are weakly increasing (co-strong) and, when reversed, are themselves an alternately co-strong sequence.

Also the number of alternately strong reversed integer partitions of n.

			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)   (311)    (51)      (61)
                    (1111)  (2111)   (222)     (322)
                            (11111)  (321)     (421)
                                     (411)     (511)
                                     (2211)    (3211)
                                     (3111)    (4111)
                                     (21111)   (22111)
                                     (111111)  (31111)
                                               (211111)
                                               (1111111)
For example, starting with the partition y = (3,2,2,1,1) and repeatedly taking run-lengths and reversing gives (3,2,2,1,1) -> (2,2,1) -> (1,2), which is not weakly decreasing, so y is not  alternately co-strong. On the other hand, we have (3,3,2,2,1,1,1) -> (3,2,2) -> (2,1) -> (1,1) -> (2) -> (1), so (3,3,2,2,1,1,1) is counted under a(13).

Cf. A000041, A100883, A181819, A182850, A182857, A304660, A305563, A317081, A317086, A317245, A317258.
The Heinz numbers of these partitions are given by A317257.
The total (instead of alternating) version is A332275.
Dominates A332289 (the normal version).
The generalization to compositions is A332338.
The dual version is A332339.
The case of reversed partitions is (also) A332339.
Cf. A316496  A317491, A329746, A332292, A332297, A332340.

tniQ[q_]:=Or[q=={},q=={1},And[LessEqual@@Length/@Split[q],tniQ[Reverse[Length/@Split[q]]]]];
Table[Length[Select[IntegerPartitions[n],tniQ]],{n,0,30}]

Updated with corrected terminology by Gus Wiseman, Mar 08 2020

A317085 Number of integer partitions of n whose sequence of multiplicities is a palindrome.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 8, 6, 11, 12, 18, 16, 31, 25, 40, 47, 60, 58, 92, 85, 125, 135, 165, 173, 248, 246, 310, 351, 435, 450, 602, 608, 766, 846, 997, 1098, 1382, 1421, 1713, 1912, 2272, 2413, 2958, 3118, 3732, 4135, 4718, 5127, 6170, 6550, 7638, 8396, 9667, 10433

Offset: 0

Gus Wiseman, Jul 21 2018

nonn

			The a(10) = 18 partitions:
(ten),
(91), (82), (73), (64), (55),
(721), (631), (541), (532),
(5221), (4411), (4321), (3322),
(33211), (32221), (22222),
(1111111111).

Chai Wah Wu, Table of n, a(n) for n = 0..166
Wikipedia, Palindrome

Cf. A000041, A000837, A025065, A124010, A242414, A317086, A317087.

Table[Length[Select[IntegerPartitions[n],Length/@Split[#]==Reverse[Length/@Split[#]]&]],{n,30}]

from sympy.utilities.iterables import partitions
def A317085(n):
    c = 1
    for d in partitions(n,m=n*2//3):
        l = len(d)
        if l > 0:
            k = sorted(d.keys())
            for i in range(l//2):
                if d[k[i]] != d[k[l-i-1]]:
                    break
            else:
                c += 1
    return c # Chai Wah Wu, Jun 22 2020

A317088 Number of normal integer partitions of n whose multiset of multiplicities is also normal.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 3, 4, 1, 4, 4, 5, 4, 6, 7, 9, 10, 13, 13, 15, 15, 17, 23, 22, 29, 29, 34, 36, 47, 45, 59, 60, 72, 77, 93, 95, 112, 121, 129, 149, 169, 176, 202, 228, 247, 268, 305, 334, 372, 405, 452, 496, 544, 594, 663, 724, 802

Offset: 0

Gus Wiseman, Jul 21 2018

nonn

A multiset is normal if it spans an initial interval of positive integers.

			The a(18) = 7 integer partitions are (543321), (5432211), (4433211), (4432221), (44322111), (4333221), (43322211).

Chai Wah Wu, Table of n, a(n) for n = 0..192

Cf. A000009, A000041, A000837, A055932, A296150, A317081, A317082, A317086.

normalQ[m_]:=Union[m]==Range[Max[m]];
Table[Length[Select[IntegerPartitions[n],And[normalQ[#],normalQ[Length/@Split[#]]]&]],{n,30}]

from sympy.utilities.iterables import partitions
from sympy import integer_nthroot
def A317088(n):
    if n == 0:
        return 1
    c = 0
    for d in partitions(n,k=integer_nthroot(2*n,2)[0]):
        l = len(d)
        if l > 0 and l == max(d):
            v = set(d.values())
            if len(v) == max(v):
                c += 1
    return c # Chai Wah Wu, Jun 23 2020

A056503 Number of periodic palindromic structures of length n using a maximum of two different symbols.

Original entry on oeis.org

1, 2, 2, 4, 4, 7, 8, 14, 16, 26, 32, 51, 64, 100, 128, 198, 256, 392, 512, 778, 1024, 1552, 2048, 3091, 4096, 6176, 8192, 12324, 16384, 24640, 32768, 49222, 65536, 98432, 131072, 196744, 262144, 393472, 524288, 786698, 1048576, 1573376, 2097152, 3146256, 4194304

Offset: 1

Marks R. Nester

nonn

For example, aaabbb is not a (finite) palindrome but it is a periodic palindrome. Permuting the symbols will not change the structure.
A periodic palindrome is just a necklace that is equivalent to its reverse. The number of binary periodic palindromes of length n is given by A164090(n). A binary periodic palindrome can only be equivalent to its complement when there are an equal number of 0's and 1's. - Andrew Howroyd, Sep 29 2017
Number of cyclic compositions (necklaces of positive integers) summing to n that can be rotated to form a palindrome. - Gus Wiseman, Sep 16 2018

			From _Gus Wiseman_, Sep 16 2018: (Start)
The sequence of palindromic cyclic compositions begins:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (111)  (22)    (113)    (33)      (115)
                    (112)   (122)    (114)     (133)
                    (1111)  (11111)  (222)     (223)
                                     (1122)    (11113)
                                     (11112)   (11212)
                                     (111111)  (11122)
                                               (1111111)
(End)

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Andrew Howroyd, Table of n, a(n) for n = 1..200

Row sums of A179181.
Cf. A016116, A045674, A056508, A164090, A285012.
Cf. A000740, A000837, A008965, A025065, A059966, A242414, A296302, A317085, A317086, A317087, A318731.

(* b = A164090, c = A045674 *)
b[n_] := (1/4)*(7 - (-1)^n)*2^((1/4)*(2*n + (-1)^n - 1));
c[0] = 1; c[n_] := c[n] = If[EvenQ[n], 2^(n/2-1) + c[n/2], 2^((n-1)/2)];
a[n_?OddQ] := b[n]/2; a[n_?EvenQ] := (1/2)*(b[n] + c[n/2]);
Array[a, 45] (* Jean-François Alcover, Oct 08 2017, after Andrew Howroyd *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Function[q,And[Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And],Array[SameQ[RotateRight[q,#],Reverse[RotateRight[q,#]]]&,Length[q],1,Or]]]]],{n,15}] (* Gus Wiseman, Sep 16 2018 *)

a(2n+1) = A164090(2n+1)/2 = 2^n, a(2n) = (A164090(2n) + A045674(n))/2. - Andrew Howroyd, Sep 29 2017

a(17)-a(45) from Andrew Howroyd, Apr 07 2017

cp's OEIS Frontend

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