A325690 -id:A325690 - cp's OEIS frontend

A008642 Quarter-squares repeated.

1, 1, 2, 2, 4, 4, 6, 6, 9, 9, 12, 12, 16, 16, 20, 20, 25, 25, 30, 30, 36, 36, 42, 42, 49, 49, 56, 56, 64, 64, 72, 72, 81, 81, 90, 90, 100, 100, 110, 110, 121, 121, 132, 132, 144, 144, 156, 156, 169, 169, 182, 182, 196, 196, 210, 210, 225, 225

Offset: 0

N. J. A. Sloane

The area of the largest rectangle whose perimeter is not greater than n. - Dmitry Kamenetsky, Aug 30 2006
Also number of partitions of n into parts 1, 2 or 4. - Reinhard Zumkeller, Aug 12 2011
Let us consider a rectangle composed of unit squares. Then count how many squares are necessary to surround this rectangle by a layer whose width is 1 unit. And repeat this surrounding ad libitum. This sequence, prepended by 4 zeros and with offset 0, gives the number of rectangles that need 2*n unit squares in one of their surrounding layers. - Michel Marcus, Sep 19 2015
a(n) is the number of nonnegative integer solutions (x,y,z) for n-2 <= 2*x + 3*y + 4*z <= n. For example, the two solutions for 1 <= 2*x + 3*y + 4*z <= 3 are (1,0,0) and (0,1,0). - Ran Pan, Oct 07 2015
Conjecture: Consider the number of compositions of n>=4*k+8 into odd parts, where the order of the parts 1,3,..,2k+1 does not count. Then, as k approaches infinity, a(n-4*k-8) is equal to the number of these restricted compositions minus A000009(n), the number of strict partitions of n. - Gregory L. Simay, Aug 12 2016
From Gus Wiseman, May 17 2019: (Start)
Also the number of length-3 integer partitions of n + 4 whose largest part is greater than the sum of the other two. These are unordered triples that cannot be the sides of a triangle. For example, the a(1) = 1 through a(10) = 9 partitions are (A = 10, B = 11, C = 12):
  (311)  (411)  (421)  (521)  (522)  (622)  (632)  (732)   (733)   (833)
                (511)  (611)  (531)  (631)  (641)  (741)   (742)   (842)
                              (621)  (721)  (722)  (822)   (751)   (851)
                              (711)  (811)  (731)  (831)   (832)   (932)
                                            (821)  (921)   (841)   (941)
                                            (911)  (A11)   (922)   (A22)
                                                           (931)   (A31)
                                                           (A21)   (B21)
                                                           (B11)   (C11)
(End)
This sequence, prepended by four 0's and with offset 0, is the number of partitions of n into four parts whose smallest two parts are equal. - Wesley Ivan Hurt, Jan 05 2021
This sequence, prepended by four 0's and with offset 0, is the number of incongruent obtuse triangles formed from the vertices of a regular n-gon. - Frank M Jackson, Nov 27 2022

D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 105.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 112, D(n).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Sara C. Billey, Matjaž Konvalinka, and Joshua P. Swanson, Asymptotic normality of the major index on standard tableaux, arXiv:1905.00975 [math.CO], 2019.
Ran Pan, Exercise U, Project P.
Harold N. Ward, A Normal Graph Algebra, arXiv:2201.00389 [math.CO], 2022.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1,1,-1,-1,1).

Cf. A002620.

Cf. A001399, A005044 (triangles without self-intersections), A069905, A124278, A266223, A325686, A325689, A325690, A325691, A325695.

[Floor(((n+1)*((-1)^n+n+6)+9)/16): n in [0..70]]; // Vincenzo Librandi, Apr 02 2014

seq((7/8+(-1)^k/8 + k + k^2/4)$2, k=0..100); # Robert Israel, Oct 08 2015

CoefficientList[Series[1/((1-x)(1-x^2)(1-x^4)), {x, 0, 70}], x] (* Vincenzo Librandi, Apr 02 2014 *)
LinearRecurrence[{1,1,-1,1,-1,-1,1},{1,1,2,2,4,4,6}, 70] (* Harvey P. Dale, Jun 03 2015 *)
Table[Floor[((n + 1) ((-1)^n + n + 6) + 9)/16], {n, 0, 70}] (* Michael De Vlieger, Aug 14 2016 *)

Vec(1/((1-x)*(1-x^2)*(1-x^4)) + O(x^70)) \\ Michel Marcus, Mar 31 2014

vector(70, n, n--; floor(((n+1)*((-1)^n+n+6)+9)/16)) \\ Altug Alkan, Oct 08 2015

[floor(floor(n/2+2)^2/2)/2 for n in (0..70)] # Bruno Berselli, Mar 03 2016

G.f.: 1/((1-x)*(1-x^2)*(1-x^4)).
a(n) = (2*n^2 + 14*n + 21 + (2*n + 7)*(-1)^n)/32 + ((1 + (-1)^n)/2 - (1 - (-1)^n)*i/2)*i^n/8, with i = sqrt(-1).
a(n) = floor(((n+1)*((-1)^n+n+6)+9)/16). - Tani Akinari, Jun 16 2013
a(n) = Sum_{i=1..floor((n+6)/2)} floor((n+6-2*i-(n mod 2))/4). - Wesley Ivan Hurt, Mar 31 2014
a(0)=1, a(1)=1, a(2)=2, a(3)=2, a(4)=4, a(5)=4, a(6)=6; for n>6, a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) - a(n-5) - a(n-6) + a(n-7). - Harvey P. Dale, Jun 03 2015
a(n) = floor(floor(n/2+2)^2/4) = floor(floor(n/2+2)^2/2)/2. - Bruno Berselli, Mar 03 2016
E.g.f.: ((14 + 7*x + x^2)*cosh(x) + 2*(cos(x) + sin(x)) + (7 + 9*x + x^2)*sinh(x))/16. - Stefano Spezia, Mar 05 2023
a(n) = floor((n + 4)/4)*floor((n + 6)/4). - Ridouane Oudra, Apr 01 2023

A325691 Number of length-3 integer partitions of n whose largest part is not greater than the sum of the other two.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 2, 3, 3, 4, 4, 6, 5, 7, 7, 9, 8, 11, 10, 13, 12, 15, 14, 18, 16, 20, 19, 23, 21, 26, 24, 29, 27, 32, 30, 36, 33, 39, 37, 43, 40, 47, 44, 51, 48, 55, 52, 60, 56, 64, 61, 69, 65, 74, 70, 79, 75, 84, 80, 90, 85, 95, 91, 101, 96, 107, 102, 113

Offset: 0

Gus Wiseman, May 15 2019

Also the number of possible triples of edge-lengths of a triangle with perimeter n, where degenerate (self-intersecting) triangles are allowed.

The number of triples (a,b,c) for 1 <= a <= b <= c <= a+b and a+b+c = n. - Yuchun Ji, Oct 15 2020

			The a(3) = 1 through a(12) = 6 partitions:
  (111)  (211)  (221)  (222)  (322)  (332)  (333)  (433)  (443)  (444)
                       (321)  (331)  (422)  (432)  (442)  (533)  (543)
                                     (431)  (441)  (532)  (542)  (552)
                                                   (541)  (551)  (633)
                                                                 (642)
                                                                 (651)

Stefano Spezia, Table of n, a(n) for n = 0..10000
Colin Defant, Michael Joseph, Matthew Macauley, and Alex McDonough, Torsors and tilings from toric toggling, arXiv:2305.07627 [math.CO], 2023. See g.f. at p. 20.
Index entries for linear recurrences with constant coefficients, signature (0,1,1,1,-1,-1,-1,0,1).

Cf. A001399, A005044 (nondegenerate triangles), A008642, A069905, A124278.

Cf. A325688, A325690, A325694.

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

Conjectures from Colin Barker, May 16 2019: (Start)
G.f.: x^3*(1 + x - x^4) / ((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>8. (End)
a(n) = A005044(n+3) - A000035(n+3). i.e., remove the only one triple (a=0,b,b) if n is even from the A005044 which is the number of triples (a,b,c) for 0 <= a <= b <= c <= a+b and a+b+c = n. - Yuchun Ji, Oct 15 2020
The above conjectured formulas are true. - Stefano Spezia, May 19 2023

A325592 Triangle read by rows where T(n,k) is the number of length-k knapsack partitions of n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 0, 1, 0, 1, 2, 2, 0, 1, 0, 1, 3, 2, 0, 0, 1, 0, 1, 3, 4, 2, 0, 0, 1, 0, 1, 4, 3, 3, 0, 0, 0, 1, 0, 1, 4, 7, 2, 2, 0, 0, 0, 1, 0, 1, 5, 6, 4, 2, 0, 0, 0, 0, 1, 0, 1, 5, 10, 6, 4, 2, 0, 0, 0, 0, 1, 0, 1, 6, 9, 5, 1, 2, 0, 0, 0, 0, 0, 1

Offset: 0

Gus Wiseman, May 15 2019

A knapsack partition of n is an integer partition of n whose distinct submultisets all have different sums.

			Triangle begins:
  1
  0  1
  0  1  1
  0  1  1  1
  0  1  2  0  1
  0  1  2  2  0  1
  0  1  3  2  0  0  1
  0  1  3  4  2  0  0  1
  0  1  4  3  3  0  0  0  1
  0  1  4  7  2  2  0  0  0  1
  0  1  5  6  4  2  0  0  0  0  1
  0  1  5 10  6  4  2  0  0  0  0  1
  0  1  6  9  5  1  2  0  0  0  0  0  1
  0  1  6 14 10  5  2  2  0  0  0  0  0  1
  0  1  7 13 11  3  3  2  0  0  0  0  0  0  1
  0  1  7 19 16  7  3  2  2  0  0  0  0  0  0  1
Row n = 12 counts the following partitions (A = 10, B = 11, C = 12):
   (C)  (66)   (444)   (3333)  (81111)  (222222)  (111111111111)
        (75)   (543)   (5511)           (711111)
        (84)   (552)   (7221)
        (93)   (732)   (7311)
        (A2)   (741)   (9111)
        (B1)   (822)
               (831)
               (921)
               (A11)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..10010

Row sums are A000041.
Column k = 2 is A004526.
Column k = 3 is A325690.
Cf. A002219, A006827, A108917, A143823, A169942, A275972, A276024, A292886, A321143, A325676, A325687.

Table[Length[Select[IntegerPartitions[n,{k}],UnsameQ@@Total/@Union[Subsets[#]]&]],{n,0,15},{k,0,n}]

A325688 Number of length-3 compositions of n such that every distinct consecutive subsequence has a different sum.

Original entry on oeis.org

0, 0, 0, 1, 0, 4, 5, 12, 12, 25, 24, 40, 41, 60, 60, 85, 84, 112, 113, 144, 144, 181, 180, 220, 221, 264, 264, 313, 312, 364, 365, 420, 420, 481, 480, 544, 545, 612, 612, 685, 684, 760, 761, 840, 840, 925, 924, 1012, 1013, 1104, 1104, 1201, 1200, 1300, 1301, 1404

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 13 2022

			The a(3) = 1 through a(8) = 12 compositions:
  (111)  (113)  (114)  (115)  (116)
         (122)  (132)  (124)  (125)
         (221)  (222)  (133)  (143)
         (311)  (231)  (142)  (152)
                (411)  (214)  (215)
                       (223)  (233)
                       (241)  (251)
                       (322)  (332)
                       (331)  (341)
                       (412)  (512)
                       (421)  (521)
                       (511)  (611)

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

Column k = 3 of A325687.
Cf. A000217 (all length-3).
Cf. A000079, A001399, A005044, A008642, A069905, A108917, A124278, A266223.
Cf. A325686, A325689, A325690, A325691.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{3}],UnsameQ@@Total/@Union[ReplaceList[#,{_,s__,_}:>{s}]]&]],{n,0,30}]

Conjectures from Colin Barker, May 16 2019: (Start)
G.f.: x^3*(1 + 2*x^2 + 4*x^3 + 5*x^4) / ((1 - x)^3*(1 + x)^2*(1 + x + x^2)).
a(n) = 2*a(n-2) + a(n-3) - a(n-4) - 2*a(n-5) + a(n-7) for n>7.
(End)

A325686 Number of strict length-3 compositions x + y + z = n satisfying x + y != z and x != y + z.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 2, 6, 8, 18, 16, 30, 34, 48, 48, 72, 72, 96, 98, 126, 128, 162, 160, 198, 202, 240, 240, 288, 288, 336, 338, 390, 392, 450, 448, 510, 514, 576, 576, 648, 648, 720, 722, 798, 800, 882, 880, 966, 970, 1056, 1056, 1152, 1152, 1248, 1250, 1350, 1352

Offset: 0

Gus Wiseman, May 13 2019

nonn

A strict composition of n is a finite sequence of distinct positive integers summing to n.
From Kevin O'Bryant, Jun 02 2025: (Start)
Also the number of Sidon sets in {0,1,...,n} with 4 elements that contain both 0 and n.
Also, the number of 3-tuples of positive integers with the 6 numbers x, y, z, x+y, y+z, x+y+z=n all distinct. (End)

			The a(6) = 2 through a(10) = 16 compositions:
  (132)  (124)  (125)  (126)  (127)
  (231)  (142)  (143)  (135)  (136)
         (214)  (152)  (153)  (154)
         (241)  (215)  (162)  (163)
         (412)  (251)  (216)  (172)
         (421)  (341)  (234)  (217)
                (512)  (243)  (253)
                (521)  (261)  (271)
                       (315)  (316)
                       (324)  (352)
                       (342)  (361)
                       (351)  (451)
                       (423)  (613)
                       (432)  (631)
                       (513)  (712)
                       (531)  (721)
                       (612)
                       (621)

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

Column k = 3 of A325677.
Cf. A000079, A001399, A005044, A108917, A124278, A143823, A266223, A275972.
Cf. A325676, A325688, A325689, A325690, A325691.

Table[Length[Cases[Join@@Permutations/@IntegerPartitions[n,{3}],{x_,y_,z_}/;x!=y!=z&&x+y!=z &&x!=y+z]],{n,0,30}]

Conjectures from Colin Barker, May 14 2019: (Start)
G.f.: 2*x^6*(1 + 3*x + 3*x^2 + 5*x^3) / ((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)
Above conjecture confirmed for n <= 5000. - Fausto A. C. Cariboni, Feb 17 2022

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)

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

cp's OEIS Frontend

A008642 Quarter-squares repeated.

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A325691 Number of length-3 integer partitions of n whose largest part is not greater than the sum of the other two.

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A325592 Triangle read by rows where T(n,k) is the number of length-k knapsack partitions of n.

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A325688 Number of length-3 compositions of n such that every distinct consecutive subsequence has a different sum.

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A325686 Number of strict length-3 compositions x + y + z = n satisfying x + y != z and x != y + z.

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

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