A325850 -id:A325850 - cp's OEIS frontend

A238424 Number of partitions of n without three consecutive parts in arithmetic progression.

1, 1, 2, 2, 4, 5, 6, 8, 13, 13, 19, 24, 30, 36, 47, 54, 72, 85, 106, 123, 151, 178, 220, 256, 314, 362, 432, 505, 605, 692, 827, 953, 1121, 1303, 1522, 1729, 2037, 2321, 2691, 3095, 3577, 4061, 4699, 5334, 6126, 6959, 7966, 9005, 10317, 11638, 13252, 14977

Offset: 0

Joerg Arndt and Alois P. Heinz, Feb 26 2014

nonn

Also the number of partitions of n whose first differences are an anti-run, meaning there are no adjacent equal differences. - Gus Wiseman, Mar 31 2020

			The a(8) = 13 such partitions are:
01:  [ 3 2 2 1 ]
02:  [ 3 3 1 1 ]
03:  [ 3 3 2 ]
04:  [ 4 2 1 1 ]
05:  [ 4 2 2 ]
06:  [ 4 3 1 ]
07:  [ 4 4 ]
08:  [ 5 2 1 ]
09:  [ 5 3 ]
10:  [ 6 1 1 ]
11:  [ 6 2 ]
12:  [ 7 1 ]
13:  [ 8 ]

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000 (terms 0..300 from Joerg Arndt and Alois P. Heinz, terms 301..350 from Fausto A. C. Cariboni)
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts

Cf. A238433 (partitions avoiding equidistant arithmetic progressions).
Cf. A238571 (partitions avoiding any arithmetic progression).
Cf. A238687.
The version for compositions is A238423, with strict case A325849.
The version for permutations is A295370.
The strict case is A332668.
The Heinz numbers of these partitions are the complement of A333195.
Partitions with equal differences are A049988.
Cf. A007862, A175342, A325850, A325851, A325852, A325874, A333631.

a[n_,r_,d_] := a[n,r,d] = Block[{j}, If[n == 0, 1, Sum[If[j == r+d, 0, a[n-j, j, j - r]], {j, Min[n, r]}]]]; a[n_] := a[n, 2*n+1, 0]; a /@ Range[0, 100] (* Giovanni Resta, Mar 02 2014 *)
Table[Length[Select[IntegerPartitions[n],!MemberQ[Differences[#,2],0]&]],{n,0,30}] (* Gus Wiseman, Mar 31 2020 *)

A295370 Number of permutations of [n] avoiding three consecutive terms in arithmetic progression.

Original entry on oeis.org

1, 1, 2, 4, 18, 80, 482, 3280, 26244, 231148, 2320130, 25238348, 302834694, 3909539452, 54761642704, 816758411516, 13076340876500, 221396129723368, 3985720881222850, 75503196628737920, 1510373288335622576, 31634502738658957588, 696162960370556156224, 15978760340940405262668

Offset: 0

Alois P. Heinz, Nov 20 2017

nonn

These are permutations of n whose second-differences are nonzero. - Gus Wiseman, Jun 03 2019

			a(3) = 4: 132, 213, 231, 312.
a(4) = 18: 1243, 1324, 1342, 1423, 2134, 2143, 2314, 2413, 2431, 3124, 3142, 3241, 3412, 3421, 4132, 4213, 4231, 4312.

Wikipedia, Arithmetic progression
Index entries related to non-averaging sequences

Column k=0 of A295390.
Cf. A003407, A174080, A238423, A238424, A338765.
Cf. A049988, A175342, A279945, A325545, A325849, A325850, A325851, A325874, A325875.

b:= proc(s, j, k) option remember; `if`(s={}, 1,
      add(`if`(k=0 or 2*j<>i+k, b(s minus {i}, i,
          `if`(2*i-j in s, j, 0)), 0), i=s))
    end:
a:= n-> b({$1..n}, 0$2):
seq(a(n), n=0..12);

Table[Length[Select[Permutations[Range[n]],!MemberQ[Differences[#,2],0]&]],{n,0,5}] (* Gus Wiseman, Jun 03 2019 *)
b[s_, j_, k_] := b[s, j, k] = If[s == {}, 1, Sum[If[k == 0 || 2*j != i + k, b[s~Complement~{i}, i, If[MemberQ[s, 2*i - j ], j, 0]], 0], {i, s}]];
a[n_] := a[n] = b[Range[n], 0, 0];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 16}] (* Jean-François Alcover, Nov 20 2023, after Alois P. Heinz *)

a(22)-a(23) from Vaclav Kotesovec, Mar 22 2022

A325874 Number of integer partitions of n whose differences of all degrees > 1 are nonzero.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 6, 8, 12, 13, 19, 24, 26, 33, 45, 52, 66, 78, 92, 113, 129, 160, 192, 231, 268, 305, 361, 436, 501, 591, 665, 783, 897, 1071, 1228, 1361, 1593, 1834, 2101, 2452, 2685, 3129, 3526, 4067, 4568, 5189, 5868, 6655, 7565, 8468, 9400

Offset: 0

Gus Wiseman, Jun 02 2019

nonn

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2). The zeroth differences are the sequence itself, while k-th differences for k > 0 are the differences of the (k-1)-th differences. If m is the length of the sequence, its differences of all degrees are the union of the zeroth through m-th differences.

The case for all degrees including 1 is A325852.

			The a(1) = 1 through a(9) = 13 partitions:
  (1)  (2)   (3)   (4)    (5)    (6)     (7)    (8)     (9)
       (11)  (21)  (22)   (32)   (33)    (43)   (44)    (54)
                   (31)   (41)   (42)    (52)   (53)    (63)
                   (211)  (221)  (51)    (61)   (62)    (72)
                          (311)  (411)   (322)  (71)    (81)
                                 (2211)  (331)  (332)   (441)
                                         (421)  (422)   (522)
                                         (511)  (431)   (621)
                                                (521)   (711)
                                                (611)   (4221)
                                                (3221)  (4311)
                                                (3311)  (5211)
                                                        (32211)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..220
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts

Cf. A049988, A238423, A325325, A325468, A325545, A325849, A325850, A325851, A325852, A325875, A325876.

Table[Length[Select[IntegerPartitions[n],!MemberQ[Union@@Table[Differences[#,i],{i,2,Length[#]}],0]&]],{n,0,30}]

A325852 Number of (strict) integer partitions of n whose differences of all degrees are nonzero.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 5, 6, 6, 9, 11, 11, 15, 19, 19, 26, 31, 31, 41, 49, 53, 62, 75, 81, 97, 112, 124, 145, 171, 175, 215, 244, 274, 307, 344, 388, 446, 497, 561, 599, 700, 779, 881, 981, 1054, 1184, 1340, 1500, 1669, 1767, 2031, 2237, 2486, 2765, 2946, 3300

Offset: 0

Gus Wiseman, May 31 2019

nonn

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2). The zeroth differences are the sequence itself, while k-th differences for k > 0 are the differences of the (k-1)-th differences. The differences of all degrees of a sequence are the union of its zeroth through m-th differences, where m is the length of the sequence.

			The a(1) = 1 through a(11) = 11 partitions (A = 10, B = 11):
  (1)  (2)  (3)   (4)   (5)   (6)   (7)    (8)    (9)    (A)    (B)
            (21)  (31)  (32)  (42)  (43)   (53)   (54)   (64)   (65)
                        (41)  (51)  (52)   (62)   (63)   (73)   (74)
                                    (61)   (71)   (72)   (82)   (83)
                                    (421)  (431)  (81)   (91)   (92)
                                           (521)  (621)  (532)  (A1)
                                                         (541)  (542)
                                                         (631)  (632)
                                                         (721)  (641)
                                                                (731)
                                                                (821)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..250

The case for only degrees > 1 is A325874.

Cf. A049988, A175342, A238423, A279945, A295370, A325328, A325468, A325545, A325850, A325851, A325875.

Table[Length[Select[IntegerPartitions[n],!MemberQ[Union@@Table[Differences[#,i],{i,Length[#]}],0]&]],{n,0,30}]

A325851 Number of (strict) compositions of n whose differences of all degrees are nonzero.

Original entry on oeis.org

1, 1, 1, 3, 4, 7, 12, 21, 35, 58, 102, 171, 284, 485, 819, 1355, 2301, 3884, 6528, 10983, 18380, 30824, 51851

Offset: 0

Gus Wiseman, May 31 2019

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2). The zeroth differences are the sequence itself, while k-th differences for k > 0 are the differences of the (k-1)-th differences. The differences of all degrees of a sequence are the union of its zeroth through m-th differences, where m is the length of the sequence.

			The a(1) = 1 through a(7) = 21 compositions:
  (1)  (2)  (3)   (4)    (5)    (6)     (7)
            (12)  (13)   (14)   (15)    (16)
            (21)  (31)   (23)   (24)    (25)
                  (121)  (32)   (42)    (34)
                         (41)   (51)    (43)
                         (131)  (132)   (52)
                         (212)  (141)   (61)
                                (213)   (124)
                                (231)   (142)
                                (312)   (151)
                                (1212)  (214)
                                (2121)  (232)
                                        (241)
                                        (313)
                                        (412)
                                        (421)
                                        (1213)
                                        (1312)
                                        (2131)
                                        (3121)
                                        (12121)

The case for only degrees > 1 is A325875.

Cf. A049988, A175342, A238423, A295370, A325328, A325545, A325850, A325852, A325874.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MemberQ[Union@@Table[Differences[#,i],{i,Length[#]}],0]&]],{n,0,10}]

A325875 Number of compositions of n whose differences of all degrees > 1 are nonzero.

Original entry on oeis.org

1, 1, 2, 3, 7, 13, 20, 38, 69, 129, 222, 407, 726, 1313, 2318, 4146, 7432, 13296, 23759, 42458, 75714

Offset: 0

Gus Wiseman, Jun 02 2019

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2). The zeroth differences are the sequence itself, while k-th differences for k > 0 are the differences of the (k-1)-th differences. If m is the length of the sequence, its differences of all degrees are the union of the zeroth through m-th differences.
A composition of n is a finite sequence of positive integers with sum n.
The case for all degrees including 1 is A325851.

			The a(1) = 1 through a(6) = 20 compositions:
  (1)  (2)   (3)   (4)    (5)     (6)
       (11)  (12)  (13)   (14)    (15)
             (21)  (22)   (23)    (24)
                   (31)   (32)    (33)
                   (112)  (41)    (42)
                   (121)  (113)   (51)
                   (211)  (122)   (114)
                          (131)   (132)
                          (212)   (141)
                          (221)   (213)
                          (311)   (231)
                          (1121)  (312)
                          (1211)  (411)
                                  (1122)
                                  (1131)
                                  (1212)
                                  (1311)
                                  (2121)
                                  (2211)
                                  (11211)

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts

Cf. A049988, A238423, A325325, A325468, A325545, A325849, A325850, A325851, A325852, A325874, A325876.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MemberQ[Union@@Table[Differences[#,i],{i,2,Length[#]}],0]&]],{n,0,10}]

A333631 Number of permutations of {1..n} with three consecutive terms in arithmetic progression.

Original entry on oeis.org

0, 0, 0, 2, 6, 40, 238, 1760, 14076, 131732, 1308670, 14678452, 176166906, 2317481348, 32416648496, 490915956484, 7846449011500, 134291298372632, 2416652824505150, 46141903780094080, 922528719841017424, 19456439433050482412, 427837767407051523776, 9873256397944571377332

Offset: 0

Gus Wiseman, Mar 31 2020

nonn

Also permutations whose second differences have at least one zero.

			The a(3) = 2 and a(4) = 6 permutations:
  (1,2,3)  (1,2,3,4)
  (3,2,1)  (1,4,3,2)
           (2,3,4,1)
           (3,2,1,4)
           (4,1,2,3)
           (4,3,2,1)

Wikipedia, Arithmetic progression

The complement is counted by A295370.
The version for prime indices is A333195.
Strict partitions with equal differences are A049980.
Partitions with equal differences are A049988.
Compositions without triples in arithmetic progression are A238423.
Partitions without triples in arithmetic progression are A238424.
Strict partitions without triples in arithmetic progression are A332668.
Cf. A000142, A007862, A175342, A325849, A325850.

Table[Select[Permutations[Range[n]],MatchQ[Differences[#],{_,x_,x_,_}]&]//Length,{n,0,8}]

a(n) = n! - A295370(n).

a(11)-a(21) (using A295370) from Giovanni Resta, Apr 07 2020

a(22)-a(23) (using A295370) from Alois P. Heinz, Jan 27 2024

cp's OEIS Frontend

A238424 Number of partitions of n without three consecutive parts in arithmetic progression.

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A295370 Number of permutations of [n] avoiding three consecutive terms in arithmetic progression.

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A325874 Number of integer partitions of n whose differences of all degrees > 1 are nonzero.

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A325852 Number of (strict) integer partitions of n whose differences of all degrees are nonzero.

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A325851 Number of (strict) compositions of n whose differences of all degrees are nonzero.

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A325875 Number of compositions of n whose differences of all degrees > 1 are nonzero.

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A333631 Number of permutations of {1..n} with three consecutive terms in arithmetic progression.

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