A049988 -id:A049988 - cp's OEIS frontend

A342194 Number of strict compositions of n with equal differences, or strict arithmetic progressions summing to n.

1, 1, 1, 3, 3, 5, 7, 7, 7, 13, 11, 11, 17, 13, 15, 25, 17, 17, 29, 19, 23, 35, 25, 23, 39, 29, 29, 45, 33, 29, 55, 31, 35, 55, 39, 43, 65, 37, 43, 65, 51, 41, 77, 43, 51, 85, 53, 47, 85, 53, 65, 87, 61, 53, 99, 67, 67, 97, 67, 59, 119, 61, 71, 113, 75, 79, 123, 67, 79, 117

Offset: 0

Gus Wiseman, Apr 02 2021

nonn

			The a(1) = 1 through a(9) = 13 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)
                                                        (7,2)
                                                        (8,1)
                                                        (1,3,5)
                                                        (2,3,4)
                                                        (4,3,2)
                                                        (5,3,1)

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

Strict compositions in general are counted by A032020.
The unordered version is A049980.
The non-strict version is A175342.
A000203 adds up divisors.
A000726 counts partitions with alternating parts unequal.
A003242 counts anti-run compositions.
A224958 counts compositions with alternating parts unequal.
A342343 counts compositions with alternating parts strictly decreasing.
A342495 counts compositions with constant quotients.
A342527 counts compositions with alternating parts equal.
Cf. A000009, A001522, A002843, A049988, A062968, A070211, A114921, A325545, A325557, A342496, A342515, A342522.

Table[Length[Select[Join@@Permutations/@Select[IntegerPartitions[n],UnsameQ@@#&],SameQ@@Differences[#]&]],{n,0,30}]

a(n > 0) = A175342(n) - A000005(n) + 1.

a(n > 0) = 2*A049988(n) - 2*A000005(n) + 1 = 2*A049982(n) + 1.

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

A385214 Number of subsets of {1..n} without all equal lengths of maximal runs of consecutive elements increasing by 1.

Original entry on oeis.org

0, 0, 0, 0, 2, 8, 25, 66, 159, 361, 791, 1688, 3539, 7328, 15040, 30669, 62246, 125896, 253975, 511357, 1028052

Offset: 0

Gus Wiseman, Jun 25 2025

			The maximal runs of S = {1,2,4,5,6,8,9} are ((1,2),(4,5,6),(8,9)), with lengths (2,3,2), so S is counted under a(9).
The a(0) = 0 through a(5) = 8 subsets:
  .  .  .  .  {1,2,4}  {1,2,4}
              {1,3,4}  {1,2,5}
                       {1,3,4}
                       {1,4,5}
                       {2,3,5}
                       {2,4,5}
                       {1,2,3,5}
                       {1,3,4,5}

These subsets are ranked by A164708, complement A164707
The complement is counted by A243815.
For distinct instead of equal lengths we have A384176, complement A384175.
For anti-runs instead of runs we have complement of A384889, for partitions A384888.
For permutations instead of subsets we have complement of A384892, distinct A384891.
For partitions instead of subsets we have complement of A384904, strict A384886.
A034839 counts subsets by number of maximal runs, for strict partitions A116674.
A049988 counts partitions with equal run-lengths, distinct A325325.
A329738 counts compositions with equal run-lengths, distinct A329739.
A384177 counts subsets with all distinct lengths of maximal anti-runs, ranks A384879.
A384887 counts partitions with equal lengths of gapless runs, distinct A384884.
A384893 counts subsets by number of maximal anti-runs, for partitions A268193, A384905.
Cf. A010027, A044813, A164710, A383013, A384885.

Table[Length[Select[Subsets[Range[n]],!SameQ@@Length/@Split[#,#2==#1+1&]&]],{n,0,10}]

A049995 Number of arithmetic progressions of 4 or more positive integers, nondecreasing with sum <= n.

Original entry on oeis.org

0, 0, 0, 1, 2, 3, 4, 6, 7, 10, 11, 14, 15, 18, 21, 25, 26, 30, 31, 37, 40, 44, 45, 51, 55, 59, 62, 69, 70, 79, 80, 86, 89, 94, 101, 111, 112, 117, 120, 132, 133, 143, 144, 152, 162, 168, 169, 180, 184, 196, 200, 209, 210, 221, 230, 242, 246, 253, 254, 274, 275, 282, 291, 302, 312, 325, 326, 336

Offset: 1

Clark Kimberling

nonn

Sadek Bouroubi and Nevrine Benyahia Tani, Integer partitions into arithmetic progressions, Rostok. Math. Kolloq. 64 (2009), 11-16.
Sadek Bouroubi and Nesrine Benyahia Tani, Integer partitions into arithmetic progressions with an odd common difference, Integers 9(1) (2009), 77-81.
Graeme McRae, Counting arithmetic sequences whose sum is n.
Graeme McRae, Counting arithmetic sequences whose sum is n [Cached copy]
Augustine O. Munagi, Combinatorics of integer partitions in arithmetic progression, Integers 10(1) (2010), 73-82.
Augustine O. Munagi and Temba Shonhiwa, On the partitions of a number into arithmetic progressions, Journal of Integer Sequences 11 (2008), Article 08.5.4.
A. N. Pacheco Pulido, Extensiones lineales de un poset y composiciones de números multipartitos, Maestría thesis, Universidad Nacional de Colombia, 2012.
Wikipedia, Arithmetic progression.

Cf. A014405, A014406, A049980, A049981, A049982, A049983, A049987, A049988, A049989, A049990, A049991, A049992, A049993, A049994, A127938.

From Petros Hadjicostas, Sep 29 2019: (Start)
a(n) = Sum_{k = 1..n} A049994(k).
G.f.: (g.f. of A049994)/(1-x). (End)

More terms from Petros Hadjicostas, Sep 29 2019

A088043 Number of partitions of n into parts which can be arranged to form a geometric progression (possibly with a common ratio of 1). (For every partition there exists a geometric progression in which this partition fits in as successive terms.).

Original entry on oeis.org

1, 2, 3, 4, 3, 6, 4, 6, 5, 6, 3, 10, 4, 7, 8, 8, 3, 10, 3, 10, 9, 6, 3, 14, 5, 7, 7, 11, 3, 15, 5, 10, 7, 6, 8, 16, 3, 6, 8, 15, 3, 16, 4, 10, 12, 6, 3, 18, 6, 10, 7, 11, 3, 14, 7, 15, 8, 6, 3, 23, 3, 8, 14, 12, 8, 14, 3, 10, 7, 15, 3, 22, 4, 6, 12, 10, 8, 15, 3, 19, 9, 6, 3, 24, 8, 7, 7, 14, 3, 23

Offset: 1

Amarnath Murthy, Sep 20 2003

nonn

			a(15) = 8 and the partitions are (15), (5, 5, 5), (3, 3, 3, 3, 3), (1, 1, ...15 times), (1, 2, 4, 8), (5, 10), (1, 14), (3, 12).
a(31) = 5 and the partitions are (31), (1+1...,31 times), (1,5,25),(1,2,4,8,16), (1,30).

Cf. A049988.

lim = 100; A = vector(lim, i, 1); for (r = 1, lim - 1, s = r + 1; while (s <= lim, forstep (k = s, lim, s, A[k]++); s = r*s + 1)); A \\ David Wasserman, Jun 21 2005

More terms from David Wasserman, Jun 21 2005

A330285 The maximum number of arithmetic progressions in a sequence of length n.

Original entry on oeis.org

0, 0, 1, 3, 7, 12, 20, 29, 41, 55, 72, 90, 113, 137, 164, 194, 228, 263, 303, 344, 390, 439, 491, 544, 604, 666, 731, 799, 872, 946, 1027, 1109, 1196, 1286, 1379, 1475, 1579, 1684, 1792, 1903, 2021, 2140, 2266, 2393, 2525, 2662, 2802, 2943, 3093, 3245, 3402, 3562, 3727

Offset: 1

Joseph Wheat, Dec 21 2019

nonn

The partial arithmetic density D_n(A) up to n is merely the number of arithmetic progressions, A(s(n)), divided by the total number of nonempty subsets of {s(1), s(2), ..., s(n)}, i.e., A(s(n))/(2^n - 1). As n approaches infinity, D_n(A) converges to zero. Furthermore, the infinite sum of the partial densities for any sequence always converges to the total density D(A). Every infinite arithmetic progression has the same total density, Sum_{n >= 1} a(n)/(2^n - 1) = alpha ~ 1.25568880818612911696845537; sequences with a finite number of progressions have D(A) < alpha; and sequences without any arithmetic progressions have D(A) = 0.

Encyclopedia of Mathematics, Density of a sequence
Eric Weisstein's World of Mathematics, Arithmetic Progression

Cf. A049988, A130518, A104429, A065825, A092482.

Partial sums of A002541.

a(n) = sum(i=1, n, sum(j=1, i, floor((i - 1)/(j + 1))))

a(n) = Sum_{i=1..n} Sum_{j=1..i} floor((i - 1)/(j + 1)).

A342531 Triangle read by rows where T(n,k) is the number of strict integer partitions of n with maximal descent k, n >= 0, 0 <= k <= n.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Mar 25 2021

The maximal descent of an empty or singleton partition is considered to be 0.

			Triangle begins:
1
1 0
1 0 0
1 1 0 0
1 0 1 0 0
1 1 0 1 0 0
1 1 1 0 1 0 0
1 1 1 1 0 1 0 0
1 0 2 1 1 0 1 0 0
1 2 1 1 1 1 0 1 0 0
1 1 2 2 1 1 1 0 1 0 0
1 1 2 3 1 1 1 1 0 1 0 0
1 1 3 2 3 1 1 1 1 0 1 0 0
1 1 3 3 3 2 1 1 1 1 0 1 0 0
1 1 3 4 3 3 2 1 1 1 1 0 1 0 0
1 3 3 4 4 3 2 2 1 1 1 1 0 1 0 0
1 0 5 5 5 4 3 2 2 1 1 1 1 0 1 0 0
1 1 4 7 5 5 4 2 2 2 1 1 1 1 0 1 0 0
1 2 5 6 7 6 4 4 2 2 2 1 1 1 1 0 1 0 0
1 1 5 9 7 7 6 4 3 2 2 2 1 1 1 1 0 1 0 0
1 1 6 9 9 7 8 5 4 3 2 2 2 1 1 1 1 0 1 0 0
Row n = 15 counts the following strict partitions (empty columns indicated by dots, A..F = 10..15):
  F  87     753   96    762   A5   A41   B4   B31  C3  C21  D2  .  E1  .  .
     654    6432  852   843   861  9321  A32
     54321  6531  7431  951   942
                  7521  8421

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

The non-strict version is A238353.
A000041 counts partitions (strict: A000009).
A049980 counts strict partitions with equal differences.
A325325 counts partitions with distinct differences (ranking: A325368).
A325545 counts compositions with distinct differences.
Cf. A003114, A003242, A005117, A034296, A049988, A238710, A342098, A342514.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&If[Length[#]<=1,k==0,Max[Differences[Reverse[#]]]==k]&]],{n,0,15},{k,0,n}]

cp's OEIS Frontend

A342194 Number of strict compositions of n with equal differences, or strict arithmetic progressions summing to n.

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

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A385214 Number of subsets of {1..n} without all equal lengths of maximal runs of consecutive elements increasing by 1.

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Mathematica

A049995 Number of arithmetic progressions of 4 or more positive integers, nondecreasing with sum <= n.

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A088043 Number of partitions of n into parts which can be arranged to form a geometric progression (possibly with a common ratio of 1). (For every partition there exists a geometric progression in which this partition fits in as successive terms.).

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PARI

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A330285 The maximum number of arithmetic progressions in a sequence of length n.

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A342531 Triangle read by rows where T(n,k) is the number of strict integer partitions of n with maximal descent k, n >= 0, 0 <= k <= n.

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