A325695 -id:A325695 - 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

A363725 Number of integer partitions of n with a different mean, median, and mode, assuming there is a unique mode.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 3, 8, 8, 17, 19, 28, 39, 59, 68, 106, 123, 165, 220, 301, 361, 477, 605, 745, 929, 1245, 1456, 1932, 2328, 2846, 3590, 4292, 5111, 6665, 8040, 9607, 11532, 14410, 16699, 20894, 24287, 28706, 35745, 42845, 49548, 59963, 70985

Offset: 0

Gus Wiseman, Jun 22 2023

nonn

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes of {a,a,b,b,b,c,d,d,d} are {b,d}.

			The a(7) = 1 through a(13) = 17 partitions:
  (3211)  (4211)  (3321)  (5311)    (4322)    (4431)    (4432)
                  (4311)  (6211)    (4421)    (5322)    (5422)
                  (5211)  (322111)  (5411)    (6411)    (5521)
                                    (6311)    (7311)    (6322)
                                    (7211)    (8211)    (6511)
                                    (43211)   (53211)   (7411)
                                    (332111)  (432111)  (8311)
                                    (422111)  (522111)  (9211)
                                                        (54211)
                                                        (63211)
                                                        (333211)
                                                        (433111)
                                                        (442111)
                                                        (532111)
                                                        (622111)
                                                        (3322111)
                                                        (32221111)

The length-4 case appears to be A325695.
For equal instead of unequal we have A363719, ranks A363727.
Allowing multiple modes gives A363720, ranks A363730.
A000041 counts partitions, strict A000009.
A008284 counts partitions by length (or decreasing mean), strict A008289.
A359893 and A359901 count partitions by median, odd-length A359902.
A362608 counts partitions with a unique mode.
Cf. A237984, A240219, A325347, A326567/A326568, A327472, A359894, A359896, A359900, A363723, A363728.

modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n], Length[modes[#]]==1&&Mean[#]!=Median[#]!=First[modes[#]]&]],{n,0,30}]

A365545 Triangle read by rows where T(n,k) is the number of strict integer partitions of n with exactly k distinct non-subset-sums.

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 3, 0, 1, 0, 0, 1, 1, 0, 0, 3, 0, 1, 0, 0, 0, 3, 0, 0, 0, 4, 0, 1, 0, 1, 0, 0, 2, 2, 0, 0, 4, 0, 1, 0, 1, 0, 0, 0, 5, 0, 0, 0, 5, 0, 1, 0, 2, 0, 0, 0, 0, 5, 2, 0, 0, 5, 0, 1, 0

Offset: 0

Gus Wiseman, Sep 24 2023

For an integer partition y of n, we call a positive integer k <= n a non-subset-sum iff there is no submultiset of y summing to k.

Is column k = n - 7 given by A325695?

			Triangle begins:
  1
  1  0
  0  1  0
  1  0  1  0
  0  1  0  1  0
  0  0  2  0  1  0
  1  0  0  2  0  1  0
  1  0  0  0  3  0  1  0
  0  1  1  0  0  3  0  1  0
  0  0  3  0  0  0  4  0  1  0
  1  0  0  2  2  0  0  4  0  1  0
  1  0  0  0  5  0  0  0  5  0  1  0
  2  0  0  0  0  5  2  0  0  5  0  1  0
  2  0  1  0  0  0  8  0  0  0  6  0  1  0
  1  1  3  0  0  0  0  7  3  0  0  6  0  1  0
  2  0  4  0  1  0  0  0 12  0  0  0  7  0  1  0
  1  1  2  2  3  1  0  0  0 11  3  0  0  7  0  1  0
  2  0  3  0  7  0  1  0  0  0 16  0  0  0  8  0  1  0
  3  0  0  2  6  3  3  1  0  0  0 15  4  0  0  8  0  1  0
Row n = 12: counts the following partitions:
  (6,3,2,1)  .  .  .  .  (9,2,1)  (6,5,1)  .  .  (11,1)  .  (12)  .
  (5,4,2,1)              (8,3,1)  (6,4,2)        (10,2)
                         (7,4,1)                 (9,3)
                         (7,3,2)                 (8,4)
                         (5,4,3)                 (7,5)

Row sums are A000009, non-strict A000041.
The complement (positive subset-sums) is also A365545 with rows reversed.
Weighted row sums are A365922, non-strict A365918.
The non-strict version is A365923, complement A365658, rank stat A325799.
A046663 counts partitions without a subset summing to k, strict A365663.
A126796 counts complete partitions, ranks A325781, strict A188431.
A364350 counts combination-free strict partitions, complement A364839.
A365543 counts partitions with a submultiset summing to k, strict A365661.
A365924 counts incomplete partitions, ranks A365830, strict A365831.
Cf. A006827, A284640, A304792, A364272, A365921.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Length[Complement[Range[n], Total/@Subsets[#]]]==k&]],{n,0,10},{k,0,n}]

A325696 Number of length-3 strict compositions of n such that no part is the sum of the other two.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 6, 6, 18, 12, 30, 30, 48, 42, 72, 66, 96, 90, 126, 120, 162, 150, 198, 192, 240, 228, 288, 276, 336, 324, 390, 378, 450, 432, 510, 498, 576, 558, 648, 630, 720, 702, 798, 780, 882, 858, 966, 948, 1056, 1032, 1152, 1128, 1248, 1224, 1350

Offset: 0

Gus Wiseman, May 15 2019

nonn

A composition of n is a finite sequence of positive integers summing to n. It is strict if all parts are distinct.

			The a(6) = 6 through a(10) = 12 compositions:
  (124)  (125)  (126)  (127)
  (142)  (152)  (135)  (136)
  (214)  (215)  (153)  (163)
  (241)  (251)  (162)  (172)
  (412)  (512)  (216)  (217)
  (421)  (521)  (234)  (271)
                (243)  (316)
                (261)  (361)
                (315)  (613)
                (324)  (631)
                (342)  (712)
                (351)  (721)
                (423)
                (432)
                (513)
                (531)
                (612)
                (621)

Cf. A000079, A001399, A005044, A266223.

Cf. A325686, A325688, A325689 (non-strict case), A325695.

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

Conjectures from Colin Barker, May 16 2019: (Start)
G.f.: 6*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) = 6 * A325695(n). - Alois P. Heinz, Jun 18 2020

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A008642 Quarter-squares repeated.

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A363725 Number of integer partitions of n with a different mean, median, and mode, assuming there is a unique mode.

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A365545 Triangle read by rows where T(n,k) is the number of strict integer partitions of n with exactly k distinct non-subset-sums.

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A325696 Number of length-3 strict compositions of n such that no part is the sum of the other two.

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