A342337 -id:A342337 - cp's OEIS frontend

A342340 Number of compositions of n where each part after the first is either twice, half, or equal to the prior part.

1, 1, 2, 4, 6, 9, 17, 24, 41, 67, 109, 173, 296, 469, 781, 1284, 2109, 3450, 5713, 9349, 15422, 25351, 41720, 68590, 112982, 185753, 305752, 503041, 827819, 1361940, 2241435, 3687742, 6068537, 9985389, 16431144, 27036576, 44489533, 73205429, 120460062, 198214516, 326161107

Offset: 0

Gus Wiseman, Mar 12 2021

nonn

			The a(1) = 1 through a(6) = 17 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)
       (11)  (12)   (22)    (122)    (24)
             (21)   (112)   (212)    (33)
             (111)  (121)   (221)    (42)
                    (211)   (1112)   (222)
                    (1111)  (1121)   (1122)
                            (1211)   (1212)
                            (2111)   (1221)
                            (11111)  (2112)
                                     (2121)
                                     (2211)
                                     (11112)
                                     (11121)
                                     (11211)
                                     (12111)
                                     (21111)
                                     (111111)

Alois P. Heinz, Table of n, a(n) for n = 0..4623 (first 1001 terms from Andrew Howroyd)

The case of partitions is A342337.
The anti-run version is A342331.
A000929 counts partitions with adjacent parts x >= 2y.
A002843 counts compositions with adjacent parts x <= 2y.
A154402 counts partitions with adjacent parts x = 2y.
A224957 counts compositions with x <= 2y and y <= 2x (strict: A342342).
A274199 counts compositions with adjacent parts x < 2y.
A342094 counts partitions with adjacent x <= 2y (strict: A342095).
A342096 counts partitions without adjacent x >= 2y (strict: A342097).
A342098 counts partitions with adjacent parts x > 2y.
A342330 counts compositions with x < 2y and y < 2x (strict: A342341).
A342332 counts compositions with adjacent parts x > 2y or y > 2x.
A342333 counts compositions with adjacent parts x >= 2y or y >= 2x.
A342334 counts compositions with adjacent parts x >= 2y or y > 2x.
A342335 counts compositions with adjacent parts x >= 2y or y = 2x.
A342338 counts compositions with adjacent parts x < 2y and y <= 2x.
Cf. A000005, A003114, A003242, A034296, A167606, A342083, A342084, A342087, A342191, A342336, A342339.

b:= proc(n, i) option remember; `if`(n=0, 1, add(
      b(n-j, j), j=`if`(i=0, {$1..n}, select(x->
       x::integer and x<=n, {i/2, i, 2*i}))))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..42);  # Alois P. Heinz, May 24 2021

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],And@@Table[#[[i]]==#[[i-1]]||#[[i]]==2*#[[i-1]]||#[[i-1]]==2*#[[i]],{i,2,Length[#]}]&]],{n,0,15}]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[b[n - j, j], {j, If[i == 0, Range[n], Select[ {i/2, i, 2 i}, IntegerQ[#] && # <= n &]]}]];
a[n_] := b[n, 0];
a /@ Range[0, 42] (* Jean-François Alcover, Jun 10 2021, after Alois P. Heinz *)

seq(n)={my(M=matid(n)); for(k=1, n, for(i=1, k-1, M[i, k] = if(i%2==0, M[i/2,k-i]) + if(i*2<=k, M[i,k-i]) + if(i*3<=k, M[i*2,k-i]))); concat([1], sum(q=1, n, M[q, ]))} \\ Andrew Howroyd, Mar 13 2021

Terms a(21) and beyond from Andrew Howroyd, Mar 13 2021

A342341 Number of strict compositions of n with all adjacent parts (x, y) satisfying x < 2y and y < 2x.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 3, 3, 5, 5, 5, 9, 7, 13, 15, 17, 19, 29, 31, 39, 43, 63, 59, 75, 121, 119, 169, 167, 199, 279, 305, 343, 479, 537, 733, 789, 883, 1057, 1421, 1545, 1831, 2409, 2577, 3343, 4001, 4657, 5131, 6065, 7755, 8841, 10473, 12995, 14659, 17671, 20619, 25157, 28255, 33131, 38265, 47699, 53171, 62611, 80005, 88519, 105937, 119989

Offset: 0

Gus Wiseman, Mar 12 2021

nonn

Each quotient of adjacent parts is between 1/2 and 2 exclusive.

			The a(1) = 1 through a(17) = 17 compositions (A..G = 10..16):
  1  2  3  4  5   6  7   8   9    A    B   C    D    E     F     G
              23     34  35  45   46   47  57   58   59    69    6A
              32     43  53  54   64   56  75   67   68    78    79
                             234  235  65  345  76   86    87    97
                             432  532  74  354  85   95    96    A6
                                           435  346  347   357   358
                                           453  643  356   456   457
                                           534       653   465   475
                                           543       743   546   547
                                                     2345  564   574
                                                     2354  645   745
                                                     4532  654   754
                                                     5432  753   853
                                                           2346  2347
                                                           6432  2356
                                                                 6532
                                                                 7432

Bert Dobbelaere, Table of n, a(n) for n = 0..100

The unordered version (partitions) is A342097 (non-strict: A342096).
The non-strict version is A342330.
The version allowing equality is A342342 (non-strict: A224957).
A000929 counts partitions with adjacent parts x >= 2y.
A002843 counts compositions with adjacent parts x <= 2y.
A154402 counts partitions with adjacent parts x = 2y.
A274199 counts compositions with adjacent parts x < 2y.
A342094 counts partitions with adjacent x <= 2y (strict: A342095).
A342098 counts partitions with adjacent parts x > 2y.
A342331 counts compositions with adjacent parts x = 2y or y = 2x.
A342332 counts compositions with adjacent parts x > 2y or y > 2x.
A342333 counts compositions with adjacent parts x >= 2y or y >= 2x.
A342335 counts compositions with adjacent parts x >= 2y or y = 2x.
A342337 counts partitions with adjacent parts x = y or x = 2y.
A342338 counts compositions with adjacent parts x < 2y and y <= 2x.
Cf. A003114, A003242, A034296, A167606, A342083, A342084, A342087, A342191, A342334, A342336, A342339, A342340.

Table[Length[Select[Join@@Permutations/@Select[IntegerPartitions[n],UnsameQ@@#&],And@@Table[#[[i]]<2*#[[i-1]]&&#[[i-1]]<2*#[[i]],{i,2,Length[#]}]&]],{n,0,15}]

More terms from Bert Dobbelaere, Mar 19 2021

A342334 Number of compositions of n with all adjacent parts (x, y) satisfying x >= 2y or y > 2x.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 11, 16, 23, 35, 54, 82, 125, 193, 294, 447, 680, 1037, 1580, 2408, 3676, 5606, 8544, 13024, 19860, 30277, 46155, 70374, 107300, 163586, 249397, 380235, 579705, 883810, 1347467, 2054371, 3132102, 4775211, 7280321, 11099613, 16922503, 25800136, 39335052, 59970425, 91431195

Offset: 0

Gus Wiseman, Mar 10 2021

nonn

Also the number of compositions of n with all adjacent parts (x, y) satisfying x > 2y or y >= 2x.

			The a(1) = 1 through a(8) = 16 compositions:
  (1)  (2)  (3)   (4)   (5)    (6)    (7)     (8)
            (12)  (13)  (14)   (15)   (16)    (17)
                  (31)  (41)   (24)   (25)    (26)
                        (131)  (51)   (52)    (62)
                               (141)  (61)    (71)
                               (312)  (124)   (125)
                                      (151)   (152)
                                      (241)   (161)
                                      (313)   (251)
                                      (412)   (314)
                                      (1312)  (413)
                                              (512)
                                              (1241)
                                              (1313)
                                              (1412)
                                              (3131)

The unordered version (partitions) is A342098 or A000929 (multisets).
The version not allowing equality (i.e., strict relations) is A342332.
The version allowing equality (i.e., non-strict relations) is A342333.
Reversing operators and changing 'or' into 'and' gives A342338.
A002843 counts compositions with adjacent parts x <= 2y.
A154402 counts partitions with adjacent parts x = 2y.
A224957 counts compositions with x <= 2y and y <= 2x (strict: A342342).
A274199 counts compositions with adjacent parts x < 2y.
A342094 counts partitions with adjacent parts x <= 2y (strict: A342095).
A342096 counts partitions without adjacent x >= 2y (strict: A342097).
A342330 counts compositions with x < 2y and y < 2x (strict: A342341).
A342331 counts compositions with adjacent parts x = 2y or y = 2x.
A342335 counts compositions with adjacent parts x >= 2y or y = 2x.
A342337 counts partitions with adjacent parts x = y or x = 2y.
Cf. A003114, A003242, A034296, A167606, A342083, A342084, A342087, A342191, A342336, A342340.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],And@@Table[#[[i]]>=2*#[[i-1]]||#[[i-1]]>2*#[[i]],{i,2,Length[#]}]&]],{n,0,15}]

More terms from Joerg Arndt, Mar 12 2021

A342342 Number of strict compositions of n with all adjacent parts (x, y) satisfying x <= 2y and y <= 2x.

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 5, 5, 3, 11, 9, 11, 17, 15, 29, 39, 31, 39, 65, 57, 107, 127, 149, 155, 187, 265, 293, 419, 523, 571, 781, 763, 941, 1371, 1387, 2125, 2383, 2775, 3243, 4189, 4555, 5349, 7241, 7997, 10591, 13171, 14581, 17213, 20253, 25177, 27701, 34317

Offset: 0

Gus Wiseman, Mar 12 2021

nonn

Each quotient of adjacent parts is between 1/2 and 2 inclusive.

			The a(1) = 1 through a(12) = 17 strict compositions (A = 10, B = 11, C = 12):
  1   2   3    4   5    6     7     8    9     A      B      C
          12       23   24    34    35   36    46     47     48
          21       32   42    43    53   45    64     56     57
                        123   124        54    235    65     75
                        321   421        63    532    74     84
                                         234   1234   236    246
                                         243   1243   245    345
                                         324   3421   542    354
                                         342   4321   632    435
                                         423          1235   453
                                         432          5321   534
                                                             543
                                                             642
                                                             1236
                                                             1245
                                                             5421
                                                             6321

The non-strict version is A224957.
The case with strict relations is A342341 (non-strict: A342330).
A000929 counts partitions with adjacent parts x >= 2y.
A002843 counts compositions with adjacent parts x <= 2y.
A154402 counts partitions with adjacent parts x = 2y.
A274199 counts compositions with adjacent parts x < 2y.
A342094 counts partitions with adjacent x <= 2y (strict: A342095).
A342096 counts partitions without adjacent x >= 2y (strict: A342097).
A342098 counts partitions with adjacent parts x > 2y.
A342331 counts compositions with adjacent parts x = 2y or y = 2x.
A342332 counts compositions with adjacent parts x > 2y or y > 2x.
A342333 counts compositions with adjacent parts x >= 2y or y >= 2x.
A342335 counts compositions with adjacent parts x >= 2y or y = 2x.
A342337 counts partitions with adjacent parts x = y or x = 2y.
A342338 counts compositions with adjacent parts x < 2y and y <= 2x.
Cf. A003114, A003242, A034296, A167606, A342083, A342084, A342087, A342191, A342334, A342336, A342340.

Table[Length[Select[Join@@Permutations/@Select[IntegerPartitions[n],UnsameQ@@#&],And@@Table[#[[i]]<=2*#[[i-1]]&&#[[i-1]]<=2*#[[i]],{i,2,Length[#]}]&]],{n,0,15}]

a(40)-a(51) from Alois P. Heinz, May 24 2021

A045691 Number of binary words of length n with autocorrelation function 2^(n-1)+1.

Original entry on oeis.org

0, 1, 1, 3, 5, 11, 19, 41, 77, 159, 307, 625, 1231, 2481, 4921, 9883, 19689, 39455, 78751, 157661, 315015, 630337, 1260049, 2520723, 5040215, 10081661, 20160841, 40324163, 80643405, 161291731, 322573579, 645157041, 1290294393, 2580608475, 5161177495

Offset: 0

Torsten Sillke (torsten.sillke(AT)lhsystems.com)

nonn

From Gus Wiseman, Jan 22 2022: (Start)
Also the number of subsets of {1..n} containing n but without adjacent elements of quotient 1/2. The Heinz numbers of these sets are a subset of the squarefree terms of A320340. For example, the a(1) = 1 through a(6) = 19 subsets are:
  {1}  {2}  {3}    {4}      {5}        {6}
            {1,3}  {1,4}    {1,5}      {1,6}
            {2,3}  {3,4}    {2,5}      {2,6}
                   {1,3,4}  {3,5}      {4,6}
                   {2,3,4}  {4,5}      {5,6}
                            {1,3,5}    {1,4,6}
                            {1,4,5}    {1,5,6}
                            {2,3,5}    {2,5,6}
                            {3,4,5}    {3,4,6}
                            {1,3,4,5}  {3,5,6}
                            {2,3,4,5}  {4,5,6}
                                       {1,3,4,6}
                                       {1,3,5,6}
                                       {1,4,5,6}
                                       {2,3,4,6}
                                       {2,3,5,6}
                                       {3,4,5,6}
                                       {1,3,4,5,6}
                                       {2,3,4,5,6}
(End)

If a(n) counts subsets of {1..n} with n and without adjacent quotients 1/2:
- The version with quotients <= 1/2 is A018819, partitions A000929.
- The version with quotients < 1/2 is A040039, partitions A342098.
- The version with quotients >= 1/2 is A045690(n+1), partitions A342094.
- The version with quotients > 1/2 is A045690, partitions A342096.
- Partitions of this type are counted by A350837, ranked by A350838.
- Strict partitions of this type are counted by A350840.
- For differences instead of quotients we have A350842, strict A350844.
- Partitions not of this type are counted by A350846, ranked by A350845.
A000740 = relatively prime subsets of {1..n} containing n.
A002843 = compositions with all adjacent quotients >= 1/2.
A050291 = double-free subsets of {1..n}.
A154402 = partitions with all adjacent quotients 2.
A308546 = double-closed subsets of {1..n}, with maximum: shifted right.
A323092 = double-free integer partitions, ranked by A320340, strict A120641.
A326115 = maximal double-free subsets of {1..n}.
Cf. A000009, A001511, A003000, A003114, A116932, A274199, A323093, A342095, A342191, A342331, A342332, A342333, A342337.

Table[Length[Select[Subsets[Range[n]],MemberQ[#,n]&&And@@Table[#[[i-1]]/#[[i]]!=1/2,{i,2,Length[#]}]&]],{n,0,15}] (* Gus Wiseman, Jan 22 2022 *)

a(2*n-1) = 2*a(2*n-2) - a(n) for n >= 2; a(2*n) = 2*a(2*n-1) + a(n) for n >= 2.

More terms from Sean A. Irvine, Mar 18 2021

A364675 Number of integer partitions of n whose nonzero first differences are a submultiset of the parts.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 7, 7, 10, 12, 15, 15, 26, 25, 35, 45, 55, 60, 86, 94, 126, 150, 186, 216, 288, 328, 407, 493, 610, 699, 896, 1030, 1269, 1500, 1816, 2130, 2620, 3029, 3654, 4300, 5165, 5984, 7222, 8368, 9976, 11637, 13771, 15960, 18978, 21896, 25815, 29915

Offset: 0

Gus Wiseman, Aug 04 2023

nonn

Conjecture: For subsets of {1..n} instead of partitions of n we have A101925.

Conjecture: The strict version is A154402.

			The partition y = (3,2,1,1) has first differences (1,1,0), and (1,1) is a submultiset of y, so y is counted under a(7).
The a(1) = 1 through a(8) = 10 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (221)    (33)      (421)      (44)
             (111)  (211)   (2111)   (42)      (2221)     (422)
                    (1111)  (11111)  (222)     (3211)     (2222)
                                     (2211)    (22111)    (4211)
                                     (21111)   (211111)   (22211)
                                     (111111)  (1111111)  (32111)
                                                          (221111)
                                                          (2111111)
                                                          (11111111)

For subsets of {1..n} we appear to have A101925, A364671, A364672.
The strict case (no differences of 0) appears to be A154402.
Starting with the distinct parts gives A342337.
For disjoint multisets: A363260, subsets A364463, strict A364464.
For overlapping multisets: A364467, ranks A364537, strict A364536.
For subsets instead of submultisets we have A364673.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A236912 counts sum-free partitions, complement A237113.
A325325 counts partitions with distinct first differences.
Cf. A002865, A007862, A108917, A229816, A237667, A237668, A320347, A363225, A364272, A364345, A364466.

submultQ[cap_,fat_] := And@@Function[i,Count[fat,i] >= Count[cap,i]] /@ Union[List@@cap];
Table[Length[Select[IntegerPartitions[n], submultQ[Differences[Union[#]],#]&]], {n,0,30}]

cp's OEIS Frontend

A342340 Number of compositions of n where each part after the first is either twice, half, or equal to the prior part.

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A342341 Number of strict compositions of n with all adjacent parts (x, y) satisfying x < 2y and y < 2x.

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A342334 Number of compositions of n with all adjacent parts (x, y) satisfying x >= 2y or y > 2x.

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A342342 Number of strict compositions of n with all adjacent parts (x, y) satisfying x <= 2y and y <= 2x.

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A045691 Number of binary words of length n with autocorrelation function 2^(n-1)+1.

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A364675 Number of integer partitions of n whose nonzero first differences are a submultiset of the parts.

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