A069905 -id:A069905 - cp's OEIS frontend

A317578 Number T(n,k) of distinct integers that are product of the parts of exactly k partitions of n into 3 positive parts; triangle T(n,k), n>=3, k>=1, read by rows.

1, 1, 2, 3, 4, 5, 7, 8, 10, 12, 12, 1, 12, 2, 19, 19, 1, 22, 1, 27, 28, 1, 31, 1, 31, 3, 38, 1, 42, 1, 46, 1, 50, 1, 50, 3, 57, 2, 51, 7, 64, 3, 71, 2, 70, 5, 77, 4, 85, 3, 86, 5, 84, 9, 104, 2, 104, 5, 108, 6, 108, 8, 1, 123, 5, 122, 9, 119, 14, 136, 9, 147, 7

Offset: 3

Alois P. Heinz, Jul 31 2018

			T(13,2) = 1: only 36 is product of the parts of exactly 2 partitions of 13 into 3 positive parts: [6,6,1], [9,2,2].
T(14,2) = 2: 40 ([8,5,1], [10,2,2]) and 72 ([6,6,2], [8,3,3]).
T(39,3) = 1: 1200 ([20,15,4], [24,10,5], [25,8,6]).
T(49,3) = 2: 3024 ([24,18,7], [27,14,8], [28,12,9]) and 3600 ([20,20,9], [24,15,10], [25,12,12]).
Triangle T(n,k) begins:
   1;
   1;
   2;
   3;
   4;
   5;
   7;
   8;
  10;
  12;
  12, 1;
  12, 2;
  19;
  19, 1;
  22, 1;

Alois P. Heinz, Rows n = 3..5000, flattened

Cf. A001399, A060277, A069905, A103277, A211540.
Row sums give A306403.
Column k=1 gives A306435.

b:= proc(n) option remember; local m, c, i, j, h, w;
      m, c:= proc() 0 end, 0; forget(m);
      for i to iquo(n, 3) do for j from i to iquo(n-i, 2) do
        h:= i*j*(n-j-i);
        w:= m(h); w:= w+1; m(h):= w;
        c:= c+x^w-x^(w-1)
      od od; c
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n)):
seq(T(n), n=3..100);

b[n_] := b[n] = Module[{m, c, i, j, h, w} , m[_] = 0; c = 0; For[i = 1, i <= Quotient[n, 3], i++, For[j = i, j <= Quotient[n - i, 2], j++, h = i*j*(n-j-i); w = m[h]; w++; m[h] = w; c = c + x^w - x^(w-1)]]; c];
T[n_] := CoefficientList[b[n], x] // Rest;
T /@ Range[3, 100] // Flatten (* Jean-François Alcover, Jun 13 2021, after Alois P. Heinz *)

Sum_{k>=1} k * T(n,k) = A001399(n-3) = A069905(n) = A211540(n+2).
Sum_{k>=2} T(n,k) = A060277(n).
min { n >= 0 : T(n,k) > 0 } = A103277(k).

A325689 Number of length-3 compositions of n such that no part is the sum of the other two.

Original entry on oeis.org

0, 0, 0, 1, 0, 6, 4, 15, 12, 28, 24, 45, 40, 66, 60, 91, 84, 120, 112, 153, 144, 190, 180, 231, 220, 276, 264, 325, 312, 378, 364, 435, 420, 496, 480, 561, 544, 630, 612, 703, 684, 780, 760, 861, 840, 946, 924, 1035, 1012, 1128, 1104, 1225, 1200, 1326, 1300, 1431

Offset: 0

Gus Wiseman, May 15 2019

nonn

A composition of n is a finite sequence of positive integers summing to n.

Confirmed recurrence relation from Colin Barker for n <= 5000. - Fausto A. C. Cariboni, Feb 15 2022

			The a(3) = 1 through a(8) = 12 compositions (empty columns not shown):
  (111)  (113)  (114)  (115)  (116)
         (122)  (141)  (124)  (125)
         (131)  (222)  (133)  (152)
         (212)  (411)  (142)  (161)
         (221)         (151)  (215)
         (311)         (214)  (233)
                       (223)  (251)
                       (232)  (323)
                       (241)  (332)
                       (313)  (512)
                       (322)  (521)
                       (331)  (611)
                       (412)
                       (421)
                       (511)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..5000

Cf. A000079, A001399, A005044, A008642, A069905, A124278, A266223.

Cf. A325676, A325688, A325690, A325691, A325694.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{3}],And@@Table[#[[i]]!=Total[Delete[#,i]],{i,3}]&]],{n,0,30}]

Conjectures from Colin Barker, May 16 2019: (Start)
G.f.: x^3*(1 - x + 4*x^2) / ((1 - x)^3*(1 + x)^2) for n>5.
a(n) = -(5 + 3*(-1)^n - 2*n) * (n-2) / 4 for n>0.
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
(End)

A102214 Expansion of (1 + 4x + 4x^2)/((1+x)*(1-x)^3).

Original entry on oeis.org

1, 6, 16, 30, 49, 72, 100, 132, 169, 210, 256, 306, 361, 420, 484, 552, 625, 702, 784, 870, 961, 1056, 1156, 1260, 1369, 1482, 1600, 1722, 1849, 1980, 2116, 2256, 2401, 2550, 2704, 2862, 3025, 3192, 3364, 3540, 3721, 3906, 4096, 4290, 4489, 4692, 4900

Offset: 0

Creighton Dement, Feb 17 2005

A floretion-generated sequence.
a(n) gives the number of triples (x,y,x+y) with positive integers satisfying x < y and x + y <= 3*n. - Marcus Schmidt (marcus-schmidt(AT)gmx.net), Jan 13 2006
Number of different partitions of numbers x + y = z such that {x,y,z} are integers {1,2,3,...,3n} and z > y > x. - Artur Jasinski, Feb 09 2010
Second bisection preceded by zero is A152743. - Bruno Berselli, Oct 25 2011
a(n) has no final digit 3, 7, 8. - Paul Curtz, Mar 04 2020
One odd followed by three evens.
From Paul Curtz, Mar 06 2020: (Start)
b(n) = 0, 1, 6, 16,  30, 49, ... = 0, a(n).
(  25, 12,  4, 0,  1,  6,  16, 30, ...
  -13, -8, -4  1,  5, 10,  14, 19, ...
    5,  4,  5, 4,  5,  4,   5,  4, ... .)
b(-n) = 0, 4, 12, 25, 42, 64, 90, 121, ... .
A154589(n) are in the main diagonal of b(n) and b(-n). (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A016778, A038764, A001859, A006578, A069905, A077043, A274221, A330707.

Cf. A000326, A016789, A152743, A001651, A032766, A047237, A047251, A154589.

[(6*n*(3*n+4)+(-1)^n+7)/8: n in [0..60]]; // Vincenzo Librandi, Oct 26 2011

aa = {}; Do[i = 0; Do[Do[Do[If[x + y == z, i = i + 1], {x, y + 1, 3 n}], {y, 1, 3 n}], {z, 1, 3 n}]; AppendTo[aa, i], {n, 1, 20}]; aa (* Artur Jasinski, Feb 09 2010 *)

a(n)=(6*n*(3*n+4)+(-1)^n+7)/8 \\ Charles R Greathouse IV, Apr 16 2020

G.f.: -(4*x^2 + 4*x + 1)/((x+1)*(x-1)^3) = (1+2*x)^2/((1+x)*(1-x)^3).
a(2n) = A016778(n) = (3n+1)^2.
a(n) + a(n+1) = A038764(n+1).
a(n) = floor( (3*n+2)/2 ) * ceiling( (3*n+2)/2 ). - Marcus Schmidt (marcus-schmidt(AT)gmx.net), Jan 13 2006
a(n) = (6*n*(3*n+4) + (-1)^n+7)/8. - Bruno Berselli, Oct 25 2011
a(n) = A198392(n) + A198392(n-1). - Bruno Berselli, Nov 06 2011
From Paul Curtz, Mar 04 2020: (Start)
a(n) = A006578(n) + A001859(n) + A077043(n+1).
a(n) = A274221(2+2*n).
a(20+n) - a(n) = 30*(32+3*n).
a(1+2*n) = 3*(1+n)*(2+3*n).
a(n) = A047237(n) * A047251(n).
a(n) = A001651(n+1) * A032766(n).(End)
E.g.f.: ((4 + 21*x + 9*x^2)*cosh(x) + 3*(1 + 7*x + 3*x^2)*sinh(x))/4. - Stefano Spezia, Mar 04 2020

Definition rewritten by Bruno Berselli, Oct 25 2011

A119028 Numbers having at least 3 unique partitions into exactly 3 parts with the same product.

Original entry on oeis.org

39, 45, 49, 53, 62, 64, 65, 70, 71, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 85, 87, 88, 89, 90, 91, 92, 93, 94, 95, 97, 98, 99, 100, 101, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128

Offset: 1

Joseph Biberstine (jrbibers(AT)indiana.edu), Jul 23 2006, Aug 10 2006

nonn

That is, numbers j such that there exist positive integers a1 <= a2 <= a3, b1 <= b2 <= b3, c1 <= c2 <= c3 (unique as triples) with j = a1 + a2 + a3 = b1 + b2 + b3 = c1 + c2 + c3 and a1*a2*a3 = b1*b2*b3 = c1*c2*c3. The answer to a question raised by Tanya Khovanova, Jul 23 2006.

All integers >= 103 are members of this sequence: see second comment in A103277. - Charles Kluepfel and M. F. Hasler, Nov 23 2018

			49 = 7 + 18 + 24    7*18*24 = 3024
49 = 8 + 14 + 27    8*14*27 = 3024
49 = 9 + 12 + 28    9*12*28 = 3024
or
49 =  9 + 20 + 20   9*20*20 = 3600
49 = 10 + 15 + 24  10*15*24 = 3600
49 = 12 + 12 + 25  12*12*25 = 3600

Cf. A069905, A103277, A317578.

pdt[lst_] := lst[[1]]*lst[[2]]*lst[[3]];
tanya[n_] := Max[Length /@ Split[Sort[pdt /@ Union[ Partition[Last /@ Flatten[ FindInstance[a + b + c == n && a >= b >= c > 0, {a, b, c}, Integers,(* failsafe *) PartitionsP@n]], 3]] ]]];
Select[ Range[4, 121], tanya@# >= 3 (*or strictly = ?*) &]
Select[Range[3, 121], Max[Length /@ Split[Sort[Times @@@ Partition[Last /@ Flatten[FindInstance[a + b + c == # && a >= b >= c > 0, {a, b, c}, Integers,(* cf A069905 *) Round[ #^2/12]]], 3]]]] >= 3 &]

More terms from Robert G. Wilson v, Jul 27 2006

A049822 a(n) = 1 - tau(n) + Sum_{d|n} tau(d-1).

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 3, 2, 4, 4, 3, 4, 5, 4, 6, 5, 4, 6, 5, 6, 9, 6, 3, 6, 9, 7, 7, 8, 5, 10, 7, 6, 9, 7, 8, 11, 8, 6, 9, 10, 7, 12, 7, 8, 14, 8, 3, 10, 12, 13, 10, 11, 5, 10, 12, 12, 13, 8, 3, 14, 11, 8, 15, 11, 13, 16, 7, 9, 9, 14, 7, 14, 11, 9, 16, 12, 11, 15, 7, 14, 16, 11, 3, 18, 17, 10, 9, 12

Offset: 1

Clark Kimberling

Number of partitions of n into 3 summands 0 < a <= b <= c with b/a and c/b integers.

a(n) is the number of 1's in the n-th row of array T given by A049816. E.g., there are 5 numbers k from 1 to 13 for which the Euclidean algorithm on (13, k) has exactly 1 nonzero remainder; hence a(13) = 5.

			a(6) = 2 because of the 3 partitions of 6 into 3 parts, [4,1,1] and [2,2,2] meet the definition; [3,2,1] fails because 2 does not divide 3.
a(100) = 20 because there are 20 partitions of 100 in 3 summands 0 < a <= b <= c with integer b/a and c/b: {a, b, c} = {1, 1, 98}, {1, 3, 96}, {1, 9, 90}, {1, 11, 88}, {1, 33, 66}, {2, 2, 96}, {2, 14, 84}, {4, 4, 92}, {4, 8, 88}, {4, 12, 84}, {4, 16, 80}, {4, 24, 72}, {4, 32, 64}, {4, 48, 48}, {5, 5, 90}, {10, 10, 80}, {10, 30, 60}, {20, 20, 60}, {20, 40, 40}, {25, 25, 50}.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Column 3 of A122934.
Cf. A000005, A003238, A057427.
Cf. A069905 (number of partitions of n into 3 positive parts).

a[n_] := 1 - DivisorSigma[0, n] + DivisorSum[n, If[# == 1, 0, DivisorSigma[ 0, # - 1]]& ]; Array[a, 90] (* Jean-François Alcover, Dec 02 2015 *)

a(n) = 1 - numdiv(n) + sumdiv(n, d, if (d==1, 0, numdiv(d-1))); \\ Michel Marcus, Oct 01 2013

Additional comments from Vladeta Jovovic, Aug 23 2003, Zak Seidov, Aug 31 2006 and Franklin T. Adams-Watters, Sep 20 2006

Edited by N. J. A. Sloane, Sep 21 2006

A242771 Number of integer points in a certain quadrilateral scaled by a factor of n (another version).

Original entry on oeis.org

0, 0, 1, 3, 6, 9, 14, 19, 25, 32, 40, 48, 58, 68, 79, 91, 104, 117, 132, 147, 163, 180, 198, 216, 236, 256, 277, 299, 322, 345, 370, 395, 421, 448, 476, 504, 534, 564, 595, 627, 660, 693, 728, 763, 799, 836, 874, 912, 952, 992, 1033, 1075, 1118, 1161, 1206

Offset: 1

Michael Somos, May 22 2014

nonn

The quadrilateral is given by four vertices [(1/2, 1/3), (0, 1), (0, 0), (1, 0)] as an example on page 22 of Ehrhart 1967. Here the open line segment from (1/2, 1/3) to (0, 1) is included but the rest of the boundary is not. The sequence is denoted by d'(n).
From Gus Wiseman, Oct 18 2020: (Start)
Also the number of ordered triples of positive integers summing to n that are not strictly increasing. For example, the a(3) = 1 through a(7) = 14 triples are:
  (1,1,1)  (1,1,2)  (1,1,3)  (1,1,4)  (1,1,5)
           (1,2,1)  (1,2,2)  (1,3,2)  (1,3,3)
           (2,1,1)  (1,3,1)  (1,4,1)  (1,4,2)
                    (2,1,2)  (2,1,3)  (1,5,1)
                    (2,2,1)  (2,2,2)  (2,1,4)
                    (3,1,1)  (2,3,1)  (2,2,3)
                             (3,1,2)  (2,3,2)
                             (3,2,1)  (2,4,1)
                             (4,1,1)  (3,1,3)
                                      (3,2,2)
                                      (3,3,1)
                                      (4,1,2)
                                      (4,2,1)
                                      (5,1,1)
A001399(n-6) counts the complement (unordered strict triples).
A014311 \ A333255 ranks these compositions.
A140106 is the unordered version.
A337484 is the case not strictly decreasing either.
A337698 counts these compositions of any length, with complement A000009.
A001399(n-6) counts unordered strict triples.
A001523 counts unimodal compositions, with complement A115981.
A007318 and A097805 count compositions by length.
A069905 counts unordered triples.
A218004 counts strictly increasing or weakly decreasing compositions.
A337483 counts triples either weakly increasing or weakly decreasing.
Cf. A332834, A337461, A337481, A337482, A337604.
(End)

			G.f. = x^3 + 3*x^4 + 6*x^5 + 9*x^6 + 14*x^7 + 19*x^8 + 25*x^9 + 32*x^10 + ...

E. Ehrhart, Sur un problème de géométrie diophantienne linéaire I, (Polyèdres et réseaux), J. Reine Angew. Math. 226 1967 1-29. MR0213320 (35 #4184).
E. Ehrhart, Sur un problème de géométrie diophantienne linéaire I, (Polyèdres et réseaux), J. Reine Angew. Math. 226 1967 1-29. MR0213320 (35 #4184). [Annotated scanned copy of pages 16 and 22 only]
E. Ehrhart, Sur un problème de géométrie diophantienne linéaire II. Systemes diophantiens lineaires, J. Reine Angew. Math. 227 1967 25-49. [Annotated scanned copy of pages 47-49 only]
Wikipedia, Ehrhart polynomial
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

Cf. A002789, A010761, A147874.

Cf. A000212, A000217, A001840, A046691, A128422, A156040.

[Floor((5*n-7)*(n-1)/12): n in [1..60]]; // Vincenzo Librandi, Jun 27 2015

a[ n_] := Quotient[ 7 - 12 n + 5 n^2, 12];
a[ n_] := With[ {o = Boole[ 0 < n], c = Boole[ 0 >= n], m = Abs@n}, Length @ FindInstance[ 0 < c + x && 0 < c + y && (2 x < c + m && 4 x + 3 y < o + 3 m || m < o + 2 x && 2 x + 3 y < c + 2 m), {x, y}, Integers, 10^9]];
LinearRecurrence[{1,1,0,-1,-1,1},{0,0,1,3,6,9},90] (* Harvey P. Dale, May 28 2015 *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{3}],!Less@@#&]],{n,0,15}] (* Gus Wiseman, Oct 18 2020 *)

PARI
```
{a(n) = (7 - 12*n + 5*n^2) \ 12};
```

{a(n) = if( n<0, polcoeff( x * (2 + x^2 + x^3 + x^4) / ((1 - x)^2 * (1 - x^6)) + x * O(x^-n), -n), polcoeff( x^3 * (1 + x + x^2 + 2*x^4) / ((1 - x)^2 * (1 - x^6)) + x * O(x^n), n))};

G.f.: x^3 * (1 + 2*x + 2*x^2) / (1 - x - x^2 + x^4 + x^5 - x^6) = (x^3 + x^4 + x^5 + 2*x^7) / ((1 - x)^2 * (1 - x^6)).
a(n) = floor( A147874(n) / 12).
a(-n) = A002789(n).
a(n+1) - a(n) = A010761(n).
For n >= 6, a(n) = A000217(n-2) - A001399(n-6). - Gus Wiseman, Oct 18 2020

A309511 Number of odd parts in the partitions of n into 3 parts.

Original entry on oeis.org

0, 0, 0, 3, 2, 4, 4, 8, 8, 13, 12, 18, 18, 24, 24, 33, 32, 40, 40, 50, 50, 61, 60, 72, 72, 84, 84, 99, 98, 112, 112, 128, 128, 145, 144, 162, 162, 180, 180, 201, 200, 220, 220, 242, 242, 265, 264, 288, 288, 312, 312, 339, 338, 364, 364, 392, 392, 421, 420

Offset: 0

Wesley Ivan Hurt, Aug 05 2019

nonn

			Figure 1: The partitions of n into 3 parts for n = 3, 4, ...
                                                          1+1+8
                                                   1+1+7  1+2+7
                                                   1+2+6  1+3+6
                                            1+1+6  1+3+5  1+4+5
                                     1+1+5  1+2+5  1+4+4  2+2+6
                              1+1+4  1+2+4  1+3+4  2+2+5  2+3+5
                       1+1+3  1+2+3  1+3+3  2+2+4  2+3+4  2+4+4
         1+1+1  1+1+2  1+2+2  2+2+2  2+2+3  2+3+3  3+3+3  3+3+4    ...
-----------------------------------------------------------------------
  n  |     3      4      5      6      7      8      9     10      ...
-----------------------------------------------------------------------
a(n) |     3      2      4      4      8      8     13     12      ...
-----------------------------------------------------------------------

Ray Chandler, Table of n, a(n) for n = 0..1000
Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1,1,-1,0,0,-1,1).

Table[Sum[Sum[Mod[i, 2] + Mod[j, 2] + Mod[n - i - j, 2], {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 80}]
Table[Count[Flatten[IntegerPartitions[n,{3}]],?OddQ],{n,0,60}] (* _Harvey P. Dale, Jan 16 2022 *)

a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} ((i mod 2) + (j mod 2) + ((n-i-j) mod 2)).
From Colin Barker, Aug 06 2019: (Start)
G.f.: x^3*(3 - x + 2*x^2 + x^4 + x^5) / ((1 - x)^3*(1 + x)^2*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)).
a(n) = a(n-1) + a(n-4) - a(n-5) + a(n-6) - a(n-7) - a(n-10) + a(n-11) for n>10.
(End)
a(n) = 3*A069905(n) - A309513(n). - Ray Chandler, Mar 13 2025

A325695 Number of length-3 strict integer partitions of n such that the largest part is not the sum of the other two.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 2, 5, 5, 8, 7, 12, 11, 16, 15, 21, 20, 27, 25, 33, 32, 40, 38, 48, 46, 56, 54, 65, 63, 75, 72, 85, 83, 96, 93, 108, 105, 120, 117, 133, 130, 147, 143, 161, 158, 176, 172, 192, 188, 208, 204, 225, 221, 243, 238, 261, 257, 280, 275

Offset: 0

Gus Wiseman, May 15 2019

nonn

			The a(7) = 1 through a(15) = 12 partitions (A = 10, B = 11, C = 12):
  (421)  (521)  (432)  (631)  (542)  (543)  (643)   (653)   (654)
                (531)  (721)  (632)  (732)  (652)   (842)   (753)
                (621)         (641)  (741)  (742)   (851)   (762)
                              (731)  (831)  (751)   (932)   (843)
                              (821)  (921)  (832)   (941)   (852)
                                            (841)   (A31)   (861)
                                            (931)   (B21)   (942)
                                            (A21)           (951)
                                                            (A32)
                                                            (A41)
                                                            (B31)
                                                            (C21)

Cf. A000041, A001399, A005044, A008642, A069905, A124278.

Cf. A325686, A325690, A325691, A325694, A325696.

Table[Length[Select[IntegerPartitions[n,{3}],UnsameQ@@#&&#[[1]]!=#[[2]]+#[[3]]&]],{n,0,30}]

Conjectures from Colin Barker, May 15 2019: (Start)
G.f.: x^7*(1 + x + 2*x^2) / ((1 - x)^3*(1 + x)^2*(1 + x^2)*(1 + x + x^2)).
a(n) = a(n-2) + a(n-3) + a(n-4) - a(n-5) - a(n-6) - a(n-7) + a(n-9) for n>9.
(End)
a(n) = A325696(n)/6. - Alois P. Heinz, Jun 18 2020

A337459 Numbers k such that the k-th composition in standard order is a unimodal triple.

Original entry on oeis.org

7, 11, 13, 14, 19, 21, 25, 26, 28, 35, 37, 41, 42, 49, 50, 52, 56, 67, 69, 73, 74, 81, 82, 84, 97, 98, 100, 104, 112, 131, 133, 137, 138, 145, 146, 161, 162, 164, 168, 193, 194, 196, 200, 208, 224, 259, 261, 265, 266, 273, 274, 289, 290, 292, 321, 322, 324

Offset: 1

Gus Wiseman, Sep 07 2020

nonn

A composition of n is a finite sequence of positive integers summing to n.
A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The sequence together with the corresponding triples begins:
      7: (1,1,1)     52: (1,2,3)    133: (5,2,1)
     11: (2,1,1)     56: (1,1,4)    137: (4,3,1)
     13: (1,2,1)     67: (5,1,1)    138: (4,2,2)
     14: (1,1,2)     69: (4,2,1)    145: (3,4,1)
     19: (3,1,1)     73: (3,3,1)    146: (3,3,2)
     21: (2,2,1)     74: (3,2,2)    161: (2,5,1)
     25: (1,3,1)     81: (2,4,1)    162: (2,4,2)
     26: (1,2,2)     82: (2,3,2)    164: (2,3,3)
     28: (1,1,3)     84: (2,2,3)    168: (2,2,4)
     35: (4,1,1)     97: (1,5,1)    193: (1,6,1)
     37: (3,2,1)     98: (1,4,2)    194: (1,5,2)
     41: (2,3,1)    100: (1,3,3)    196: (1,4,3)
     42: (2,2,2)    104: (1,2,4)    200: (1,3,4)
     49: (1,4,1)    112: (1,1,5)    208: (1,2,5)
     50: (1,3,2)    131: (6,1,1)    224: (1,1,6)

Eric Weisstein's World of Mathematics, Unimodal Sequence

A337460 is the non-unimodal version.
A000217(n - 2) counts 3-part compositions.
6*A001399(n - 6) = 6*A069905(n - 3) = 6*A211540(n - 1) counts strict 3-part compositions.
A001399(n - 3) = A069905(n) = A211540(n + 2) counts 3-part partitions.
A001399(n - 6) = A069905(n - 3) = A211540(n - 1) counts strict 3-part partitions.
A001523 counts unimodal compositions.
A007052 counts unimodal patterns.
A011782 counts unimodal permutations.
A115981 counts non-unimodal compositions.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Triples are A014311, with strict case A337453.
- Sum is A070939.
- Runs are counted by A124767.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Heinz number is A333219.
- Combinatory separations are counted by A334030.
- Non-unimodal compositions are A335373.
- Non-co-unimodal compositions are A335374.
Cf. A007304, A014612, A072706, A156040, A211540, A227038, A332743, A337452, A337461, A337604.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,1000],Length[stc[#]]==3&&!MatchQ[stc[#],{x_,y_,z_}/;x>y

Complement of A335373 in A014311.

A337460 Numbers k such that the k-th composition in standard order is a non-unimodal triple.

Original entry on oeis.org

22, 38, 44, 70, 76, 88, 134, 140, 148, 152, 176, 262, 268, 276, 280, 296, 304, 352, 518, 524, 532, 536, 552, 560, 592, 608, 704, 1030, 1036, 1044, 1048, 1064, 1072, 1096, 1104, 1120, 1184, 1216, 1408, 2054, 2060, 2068, 2072, 2088, 2096, 2120, 2128, 2144, 2192

Offset: 1

Gus Wiseman, Sep 18 2020

nonn

These are triples matching the pattern (2,1,2), (3,1,2), or (2,1,3).
A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The sequence together with the corresponding triples begins:
      22: (2,1,2)     296: (3,2,4)    1048: (6,1,4)
      38: (3,1,2)     304: (3,1,5)    1064: (5,2,4)
      44: (2,1,3)     352: (2,1,6)    1072: (5,1,5)
      70: (4,1,2)     518: (7,1,2)    1096: (4,3,4)
      76: (3,1,3)     524: (6,1,3)    1104: (4,2,5)
      88: (2,1,4)     532: (5,2,3)    1120: (4,1,6)
     134: (5,1,2)     536: (5,1,4)    1184: (3,2,6)
     140: (4,1,3)     552: (4,2,4)    1216: (3,1,7)
     148: (3,2,3)     560: (4,1,5)    1408: (2,1,8)
     152: (3,1,4)     592: (3,2,5)    2054: (9,1,2)
     176: (2,1,5)     608: (3,1,6)    2060: (8,1,3)
     262: (6,1,2)     704: (2,1,7)    2068: (7,2,3)
     268: (5,1,3)    1030: (8,1,2)    2072: (7,1,4)
     276: (4,2,3)    1036: (7,1,3)    2088: (6,2,4)
     280: (4,1,4)    1044: (6,2,3)    2096: (6,1,5)

Eric Weisstein's World of Mathematics, Unimodal Sequence
Gus Wiseman, Statistics, classes, and transformations of standard compositions

A000212 counts unimodal triples.
A000217(n - 2) counts 3-part compositions.
A001399(n - 3) counts 3-part partitions.
A001399(n - 6) counts 3-part strict partitions.
A001399(n - 6)*2 counts non-unimodal 3-part strict compositions.
A001399(n - 6)*4 counts unimodal 3-part strict compositions.
A001399(n - 6)*6 counts 3-part strict compositions.
A001523 counts unimodal compositions.
A001840 counts non-unimodal triples.
A059204 counts non-unimodal permutations.
A115981 counts non-unimodal compositions.
A328509 counts non-unimodal patterns.
A337459 ranks unimodal triples.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Triples are A014311.
- Sum is A070939.
- Runs are counted by A124767.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Heinz number is A333219.
- Non-unimodal compositions are A335373.
- Non-co-unimodal compositions are A335374.
- Strict triples are A337453.
Cf. A007304, A014612, A069905, A072706, A156040, A211540, A227038, A332743, A337461, A337604.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,1000],Length[stc[#]]==3&&MatchQ[stc[#],{x_,y_,z_}/;x>y

Intersection of A014311 and A335373.

cp's OEIS Frontend

A317578 Number T(n,k) of distinct integers that are product of the parts of exactly k partitions of n into 3 positive parts; triangle T(n,k), n>=3, k>=1, read by rows.

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A325689 Number of length-3 compositions of n such that no part is the sum of the other two.

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A102214 Expansion of (1 + 4*x + 4*x^2)/((1+x)*(1-x)^3).

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A119028 Numbers having at least 3 unique partitions into exactly 3 parts with the same product.

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A049822 a(n) = 1 - tau(n) + Sum_{d|n} tau(d-1).

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A242771 Number of integer points in a certain quadrilateral scaled by a factor of n (another version).

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A309511 Number of odd parts in the partitions of n into 3 parts.

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A102214 Expansion of (1 + 4x + 4x^2)/((1+x)*(1-x)^3).