A183558 -id:A183558 - cp's OEIS frontend

A355394 Number of integer partitions of n such that, for all parts x, x - 1 or x + 1 is also a part.

1, 0, 0, 1, 1, 3, 3, 6, 6, 10, 11, 16, 18, 25, 30, 38, 47, 59, 74, 90, 112, 136, 171, 203, 253, 299, 372, 438, 536, 631, 767, 900, 1085, 1271, 1521, 1774, 2112, 2463, 2910, 3389, 3977, 4627, 5408, 6276, 7304, 8459, 9808, 11338, 13099, 15112, 17404, 20044, 23018, 26450, 30299, 34746, 39711, 45452, 51832

Offset: 0

Gus Wiseman, Aug 26 2022

nonn

These are partitions without a neighborless part, where a part x is neighborless if neither x - 1 nor x + 1 are parts. The first counted partition that does not cover an interval is (5,4,2,1).

			The a(0) = 1 through a(9) = 11 partitions:
  ()  .  .  (21)  (211)  (32)    (321)    (43)      (332)      (54)
                         (221)   (2211)   (322)     (3221)     (432)
                         (2111)  (21111)  (2221)    (22211)    (3222)
                                          (3211)    (32111)    (3321)
                                          (22111)   (221111)   (22221)
                                          (211111)  (2111111)  (32211)
                                                               (222111)
                                                               (321111)
                                                               (2211111)
                                                               (21111111)

Lucas A. Brown, Table of n, a(n) for n = 0..100
Lucas A. Brown, A355394.py

The singleton case is A355393, complement A356235.
The complement is counted by A356236, ranked by A356734.
The strict case is A356606, complement A356607.
These partitions are ranked by A356736.
A000041 counts integer partitions, strict A000009.
A000837 counts relatively prime partitions, ranked by A289509.
A007690 counts partitions with no singletons, complement A183558.
Cf. A066312, A073491, A077855, A328171, A328172, A328187, A328221, A356233, A356237.

Table[Length[Select[IntegerPartitions[n],Function[ptn,!Or@@Table[!MemberQ[ptn,x-1]&&!MemberQ[ptn,x+1],{x,Union[ptn]}]]]],{n,0,30}]

a(n) = A000041(n) - A356236(n).

a(31)-a(59) from Lucas A. Brown, Sep 04 2022

A350839 Number of integer partitions of n with a difference < -1 and a conjugate difference < -1.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 3, 7, 11, 17, 26, 39, 54, 81, 108, 148, 201, 269, 353, 467, 601, 779, 995, 1272, 1605, 2029, 2538, 3171, 3941, 4881, 6012, 7405, 9058, 11077, 13478, 16373, 19817, 23953, 28850, 34692, 41599, 49802, 59461, 70905, 84321, 100155, 118694

Offset: 0

Gus Wiseman, Jan 24 2022

nonn

We define a difference of a partition to be a difference of two adjacent parts.

			The a(5) = 1 through a(10) = 17 partitions:
  (311)  (411)   (511)    (422)     (522)      (622)
         (3111)  (4111)   (611)     (711)      (811)
                 (31111)  (3311)    (4221)     (4222)
                          (4211)    (4311)     (4411)
                          (5111)    (5211)     (5221)
                          (41111)   (6111)     (5311)
                          (311111)  (33111)    (6211)
                                    (42111)    (7111)
                                    (51111)    (42211)
                                    (411111)   (43111)
                                    (3111111)  (52111)
                                               (61111)
                                               (331111)
                                               (421111)
                                               (511111)
                                               (4111111)
                                               (31111111)

Allowing -1 gives A144300 = non-constant partitions.
Taking one of the two conditions gives A239955, ranked by A073492, A065201.
These partitions are ranked by A350841.
A000041 = integer partitions, strict A000009.
A034296 = flat (contiguous) partitions, strict A001227.
A073491 = numbers whose prime indices have no gaps, strict A137793.
A090858 = partitions with a single hole, ranked by A325284.
A116931 = partitions with differences != -1, strict A003114.
A116932 = partitions with differences != -1 or -2, strict A025157.
A277103 = partitions with the same number of odd parts as their conjugate.
A350837 = partitions with no adjacent doublings, strict A350840.
A350842 = partitions with differences != -2, strict A350844, sets A005314.
Cf. A000070, A000929, A008284, A040039, A183558, A319630, A321440, A350838.

conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],(Min@@Differences[#]<-1)&&(Min@@Differences[conj[#]]<-1)&]],{n,0,30}]

A356237 Heinz numbers of integer partitions with a neighborless singleton.

Original entry on oeis.org

2, 3, 5, 7, 10, 11, 13, 14, 17, 19, 20, 21, 22, 23, 26, 28, 29, 31, 33, 34, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 76, 78, 79, 80, 82, 83, 84, 85, 86, 87, 88, 89, 91, 92, 93

Offset: 1

Gus Wiseman, Aug 24 2022

nonn

A part x is neighborless if neither x - 1 nor x + 1 are parts, and a singleton if it appears only once.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
Also numbers that, for some prime index x, are not divisible by prime(x)^2, prime(x - 1), or prime(x + 1). Here, a prime index of n is a number m such that prime(m) divides n.

			The terms together with their prime indices begin:
   2: {1}
   3: {2}
   5: {3}
   7: {4}
  10: {1,3}
  11: {5}
  13: {6}
  14: {1,4}
  17: {7}
  19: {8}
  20: {1,1,3}
  21: {2,4}
  22: {1,5}
  23: {9}
  26: {1,6}
  28: {1,1,4}

The complement is counted by A355393.
These partitions are counted by A356235.
Not requiring a singleton gives A356734.
A001221 counts distinct prime factors, with sum A001414.
A003963 multiplies together the prime indices of n.
A007690 counts partitions with no singletons, complement A183558.
A056239 adds up prime indices, row sums of A112798, lengths A001222.
A073491 lists numbers with gapless prime indices, complement A073492.
A132747 counts non-isolated divisors, complement A132881.
A356069 counts gapless divisors, initial A356224 (complement A356225).
A356236 counts partitions with a neighborless part, complement A355394.
A356607 counts strict partitions w/ a neighborless part, complement A356606.
Cf. A286470, A289508, A325160, A328166, A328335, A356231, A356233, A356234.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Function[ptn,Or@@Table[Count[ptn,x]==1&&!MemberQ[ptn,x-1]&&!MemberQ[ptn,x+1],{x,Union[ptn]}]]@*primeMS]

A355393 Number of integer partitions of n such that, for all parts x of multiplicity 1, either x - 1 or x + 1 is also a part.

Original entry on oeis.org

1, 0, 1, 2, 3, 4, 6, 7, 10, 14, 17, 23, 32, 39, 51, 67, 83, 105, 134, 165, 206, 256, 312, 385, 475, 573, 697, 849, 1021, 1231, 1483, 1771, 2121, 2534, 3007, 3575, 4245, 5008, 5914, 6979, 8198, 9626, 11292, 13201, 15430, 18010, 20960, 24389, 28346, 32855, 38066

Offset: 0

Gus Wiseman, Aug 26 2022

nonn

These are partitions without a neighborless singleton, where a part x is neighborless if neither x - 1 nor x + 1 are parts, and a singleton if it appears only once.

			The a(0) = 1 through a(8) = 10 partitions:
  ()  .  (11)  (21)   (22)    (32)     (33)      (43)       (44)
               (111)  (211)   (221)    (222)     (322)      (332)
                      (1111)  (2111)   (321)     (2221)     (2222)
                              (11111)  (2211)    (3211)     (3221)
                                       (21111)   (22111)    (3311)
                                       (111111)  (211111)   (22211)
                                                 (1111111)  (32111)
                                                            (221111)
                                                            (2111111)
                                                            (11111111)

This is the singleton case of A355394, complement A356236.
The complement is counted by A356235.
These partitions are ranked by the complement of A356237.
The strict case is A356606, complement A356607.
A000041 counts integer partitions, strict A000009.
A000837 counts relatively prime partitions, ranked by A289509.
A007690 counts partitions with no singletons, complement A183558.
Cf. A073491, A325160, A328171, A328172, A328187, A328220, A356233.

Table[Length[Select[IntegerPartitions[n],Function[ptn,!Or@@Table[Count[ptn,x]==1&&!MemberQ[ptn,x-1]&&!MemberQ[ptn,x+1],{x,Union[ptn]}]]]],{n,0,30}]

A356235 Number of integer partitions of n with a neighborless singleton.

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 5, 8, 12, 16, 25, 33, 45, 62, 84, 109, 148, 192, 251, 325, 421, 536, 690, 870, 1100, 1385, 1739, 2161, 2697, 3334, 4121, 5071, 6228, 7609, 9303, 11308, 13732, 16629, 20101, 24206, 29140, 34957, 41882, 50060, 59745, 71124, 84598, 100365

Offset: 0

Gus Wiseman, Aug 23 2022

nonn

A part x is neighborless if neither x - 1 nor x + 1 are parts, and a singleton if it appears only once. Examples of partitions with a neighborless singleton are: (3), (3,1), (3,1,1), (3,3,1). Examples of partitions without a neighborless singleton are: (3,3,1,1), (4,3,1,1), (3,2,1), (2,1), (3,3).

			The a(1) = 1 through a(8) = 12 partitions:
  (1)  (2)  (3)  (4)   (5)    (6)     (7)      (8)
                 (31)  (41)   (42)    (52)     (53)
                       (311)  (51)    (61)     (62)
                              (411)   (331)    (71)
                              (3111)  (421)    (422)
                                      (511)    (431)
                                      (4111)   (521)
                                      (31111)  (611)
                                               (4211)
                                               (5111)
                                               (41111)
                                               (311111)

The complement is counted by A355393.
This is the singleton case of A356236, complement A355394.
These partitions are ranked by A356237.
The strict case is A356607, complement A356606.
A000041 counts integer partitions, strict A000009.
A000837 counts relatively prime partitions, ranked by A289509.
A007690 counts partitions with no singletons, complement A183558.
Cf. A066205, A325160, A328171, A328172, A328187, A328221, A356233.

Table[Length[Select[IntegerPartitions[n],Min@@Length/@Split[Reverse[#],#1>=#2-1&]==1&]],{n,0,30}]

A356236 Number of integer partitions of n with a neighborless part.

Original entry on oeis.org

0, 1, 2, 2, 4, 4, 8, 9, 16, 20, 31, 40, 59, 76, 105, 138, 184, 238, 311, 400, 515, 656, 831, 1052, 1322, 1659, 2064, 2572, 3182, 3934, 4837, 5942, 7264, 8872, 10789, 13109, 15865, 19174, 23105, 27796, 33361, 39956, 47766, 56985, 67871, 80675, 95750, 113416

Offset: 0

Gus Wiseman, Aug 24 2022

nonn

A part x of a partition is neighborless if neither x - 1 nor x + 1 are parts.

			The a(1) = 1 through a(8) = 9 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (111)  (22)    (41)     (33)      (52)
                    (31)    (311)    (42)      (61)
                    (1111)  (11111)  (51)      (331)
                                     (222)     (421)
                                     (411)     (511)
                                     (3111)    (4111)
                                     (111111)  (31111)
                                               (1111111)

The complement is counted by A355394, singleton case A355393.
The singleton case is A356235, ranked by A356237.
The strict case is A356607, complement A356606.
These partitions are ranked by the complement of A356736.
A000041 counts integer partitions, strict A000009.
A000837 counts relatively prime partitions, ranked by A289509.
A007690 counts partitions with no singletons, complement A183558.
Cf. A066205, A112798, A319630, A325160, A328171, A328172, A328187, A328221.

Table[Length[Select[IntegerPartitions[n],Function[ptn,Or@@Table[!MemberQ[ptn,x-1]&&!MemberQ[ptn,x+1],{x,Union[ptn]}]]]],{n,0,30}]

a(n) = A000041(n) - A355394(n).

A183563 Number of partitions of n containing a clique of size 6.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 5, 5, 8, 10, 15, 18, 27, 33, 47, 57, 78, 96, 129, 159, 208, 258, 330, 407, 517, 635, 798, 978, 1217, 1482, 1833, 2225, 2729, 3303, 4028, 4856, 5885, 7070, 8528, 10211, 12259, 14628, 17494, 20800, 24777, 29378, 34867

Offset: 6

Alois P. Heinz, Jan 05 2011

nonn

All parts of a number partition with the same value form a clique. The size of a clique is the number of elements in the clique.

			a(10) = 2, because 2 partitions of 10 contain (at least) one clique of size 6: [1,1,1,1,1,1,2,2], [1,1,1,1,1,1,4].

Alois P. Heinz, Table of n, a(n) for n = 6..1000

6th column of A183568. Cf. A000041, A183558, A183559, A183560, A183561, A183562, A183564, A183565, A183566, A183567.

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->`if`(j=6, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> (l-> l[2])(b(n, n)):
seq(a(n), n=6..55);

max = 55; f = (1 - Product[1 - x^(6j) + x^(7j), {j, 1, max}])/Product[1 - x^j, {j, 1, max}]; s = Series[f, {x, 0, max}]; Drop[CoefficientList[s, x], 6] (* Jean-François Alcover, Oct 01 2014 *)

G.f.: (1-Product_{j>0} (1-x^(6*j)+x^(7*j))) / (Product_{j>0} (1-x^j)).

a(n) = A000041(n) - A184641(n). - Vaclav Kotesovec, Jun 12 2025

A127002 Number of partitions of n that have the form a+a+b+c where a,b,c are distinct.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 2, 4, 3, 7, 8, 11, 11, 17, 17, 23, 23, 30, 31, 39, 38, 48, 49, 58, 58, 70, 70, 82, 82, 95, 96, 110, 109, 125, 126, 141, 141, 159, 159, 177, 177, 196, 197, 217, 216, 238, 239, 260, 260, 284, 284, 308, 308, 333, 334, 360, 359, 387, 388, 415, 415, 445

Offset: 1

Clark Kimberling, Jan 01 2007

From Gus Wiseman, Apr 19 2019: (Start)
Also the number of integer partitions of n - 4 of the form a+b, a+a+b, or a+a+b+c, ignoring ordering. A bijection can be constructed from the partitions described in the name by subtracting one from all parts and deleting zeros. These are also partitions with adjusted frequency depth (A323014, A325280) equal to their length plus one, and their Heinz numbers are given by A325281. For example, the a(7) = 1 through a(13) = 11 partitions are:
  (21)  (31)   (32)   (42)   (43)    (53)    (54)
        (211)  (41)   (51)   (52)    (62)    (63)
               (221)  (411)  (61)    (71)    (72)
               (311)         (322)   (332)   (81)
                             (331)   (422)   (441)
                             (511)   (611)   (522)
                             (3211)  (3221)  (711)
                                     (4211)  (3321)
                                             (4221)
                                             (4311)
                                             (5211)
(End)

			a(10) counts these partitions: {1,1,2,6}, (1,1,3,5), {2,2,1,5}.
a(11) counts {1,1,2,7}, {1,1,3,6}, {1,1,4,5}, {2,2,1,6}, {2,2,3,4}, {3,3,1,4}, {4,4,1,2}
From _Gus Wiseman_, Apr 19 2019: (Start)
The a(7) = 1 through a(13) = 11 partitions of the form a+a+b+c are the following. The Heinz numbers of these partitions are given by A085987.
  (3211)  (3221)  (3321)  (5221)  (4322)  (4332)  (4432)
          (4211)  (4221)  (5311)  (4331)  (4431)  (5332)
                  (4311)  (6211)  (4421)  (5322)  (5422)
                  (5211)          (5411)  (5331)  (5521)
                                  (6221)  (6411)  (6322)
                                  (6311)  (7221)  (6331)
                                  (7211)  (7311)  (6511)
                                          (8211)  (7411)
                                                  (8221)
                                                  (8311)
                                                  (9211)
(End)

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,1,1,1,-1,-1,-1,0,1)

Cf. A000041, A008284, A085987, A090858, A116608, A117571, A183558, A325242, A325244, A325280, A325281.

R:=PowerSeriesRing(Integers(), 70); [0,0,0,0,0,0] cat Coefficients(R!( x^7*(1+2*x+3*x^2)/((1-x^2)*(1-x^3)*(1-x^4)) )); // G. C. Greubel, May 30 2019

g:=sum(sum(sum(x^(i+j+k)*(x^i+x^j+x^k),i=1..j-1),j=2..k-1),k=3..80): gser:=series(g,x=0,70): seq(coeff(gser,x,n),n=1..65); # Emeric Deutsch, Jan 05 2007
isA127002 := proc(p) local s; if nops(p) = 4 then s := convert(p,set) ; if nops(s) = 3 then RETURN(1) ; else RETURN(0) ; fi ; else RETURN(0) ; fi ; end:
A127002 := proc(n) local part,res,p; part := combinat[partition](n) ; res := 0 ; for p from 1 to nops(part) do res := res+isA127002(op(p,part)) ; od ; RETURN(res) ; end:
for n from 1 to 200 do print(A127002(n)) ; od ; # R. J. Mathar, Jan 07 2007

Table[Length[Select[IntegerPartitions[n],Sort[Length/@Split[#]]=={1,1,2}&]],{n,70}] (* Gus Wiseman, Apr 19 2019 *)
Rest[CoefficientList[Series[x^7*(1+2*x+3*x^2)/((1-x^2)*(1-x^3)*(1-x^4)), {x,0,70}], x]] (* G. C. Greubel, May 30 2019 *)

my(x='x+O('x^70)); concat(vector(6), Vec(x^7*(1+2*x+3*x^2)/((1-x^2)*(1-x^3)*(1-x^4)))) \\ G. C. Greubel, May 30 2019

a=(x^7*(1+2*x+3*x^2)/((1-x^2)*(1-x^3)*(1-x^4))).series(x, 70).coefficients(x, sparse=False); a[1:] # G. C. Greubel, May 30 2019

G.f.: x^7*(1+2*x+3*x^2)/((1-x^2)*(1-x^3)*(1-x^4)) - Vladeta Jovovic, Jan 03 2007
G.f.: Sum_{k>=3} Sum_{j=2..k-1} Sum_{m=1..j-1} x^(m+j+k)*(x^m +x^j +x^k). - Emeric Deutsch, Jan 05 2007
a(n) = binomial(floor((n-1)/2),2) - floor((n-1)/3) - floor((n-1)/4) + floor(n/4). - Mircea Merca, Nov 23 2013
a(n) = A005044(n-4) + 2*A005044(n-3) + 3*A005044(n-2). - R. J. Mathar, Nov 23 2013

A183559 Number of partitions of n containing a clique of size 2.

Original entry on oeis.org

1, 0, 2, 2, 3, 5, 9, 10, 16, 23, 31, 43, 60, 75, 106, 140, 179, 237, 310, 389, 508, 647, 815, 1032, 1305, 1617, 2033, 2527, 3117, 3857, 4764, 5812, 7142, 8711, 10585, 12866, 15605, 18803, 22716, 27325, 32774, 39286, 47016, 56019, 66819, 79456, 94273, 111766

Offset: 2

Alois P. Heinz, Jan 05 2011

nonn

All parts of a number partition with the same value form a clique. The size of a clique is the number of elements in the clique.

			a(7) = 5, because 5 partitions of 7 contain (at least) one clique of size 2: [1,1,1,2,2], [1,1,2,3], [2,2,3], [1,3,3], [1,1,5].

Alois P. Heinz, Table of n, a(n) for n = 2..1000

Column k=2 of A183568.

Cf. A000041, A183558, A183560, A183561, A183562, A183563, A183564, A183565, A183566, A183567.

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->`if`(j=2, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> (l-> l[2])(b(n, n)):
seq(a(n), n=2..50);

max = 50; f = (1 - Product[1 - x^(2j) + x^(3j), {j, 1, max}])/Product[1 - x^j, {j, 1, max}]; s = Series[f, {x, 0, max}]; Drop[CoefficientList[s, x], 2] (* Jean-François Alcover, Oct 01 2014 *)

G.f.: (1-Product_{j>0} (1-x^(2*j)+x^(3*j))) / (Product_{j>0} (1-x^j)).

a(n) = A000041(n) - A116645(n). - Vaclav Kotesovec, Jun 12 2025

A183560 Number of partitions of n containing a clique of size 3.

Original entry on oeis.org

1, 0, 1, 2, 3, 3, 6, 8, 13, 15, 24, 30, 44, 54, 77, 98, 134, 165, 222, 279, 367, 454, 588, 731, 936, 1148, 1454, 1788, 2241, 2732, 3400, 4140, 5106, 6183, 7579, 9157, 11156, 13406, 16249, 19482, 23489, 28042, 33666, 40087, 47914, 56851

Offset: 3

Alois P. Heinz, Jan 05 2011

nonn

All parts of a number partition with the same value form a clique. The size of a clique is the number of elements in the clique.

			a(9) = 6, because 6 partitions of 9 contain (at least) one clique of size 3: [1,1,1,2,2,2], [2,2,2,3], [1,1,1,3,3], [3,3,3], [1,1,1,2,4], [1,1,1,6].

Alois P. Heinz, Table of n, a(n) for n = 3..1000

Column k=3 of A183568.

Cf. A000041, A183558, A183559, A183561, A183562, A183563, A183564, A183565, A183566, A183567.

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->`if`(j=3, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> (l-> l[2])(b(n, n)):
seq(a(n), n=3..50);

max = 50; f = (1 - Product[1 - x^(3j) + x^(4j), {j, 1, max}])/Product[1 - x^j, {j, 1, max}]; s = Series[f, {x, 0, max}]; Drop[CoefficientList[s, x], 3] (* Jean-François Alcover, Oct 01 2014 *)

G.f.: (1-Product_{j>0} (1-x^(3*j)+x^(4*j))) / (Product_{j>0} (1-x^j)).

a(n) = A000041(n) - A118807(n). - Vaclav Kotesovec, Jun 12 2025

cp's OEIS Frontend

A355394 Number of integer partitions of n such that, for all parts x, x - 1 or x + 1 is also a part.

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A350839 Number of integer partitions of n with a difference < -1 and a conjugate difference < -1.

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A356237 Heinz numbers of integer partitions with a neighborless singleton.

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Mathematica

A355393 Number of integer partitions of n such that, for all parts x of multiplicity 1, either x - 1 or x + 1 is also a part.

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Mathematica

A356235 Number of integer partitions of n with a neighborless singleton.

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Mathematica

A356236 Number of integer partitions of n with a neighborless part.

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Mathematica

Formula

A183563 Number of partitions of n containing a clique of size 6.

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Maple

Mathematica

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A127002 Number of partitions of n that have the form a+a+b+c where a,b,c are distinct.

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