A077043 -id:A077043 - cp's OEIS frontend

A256315 Number of partitions of 3n into exactly 6 parts.

0, 0, 1, 3, 11, 26, 58, 110, 199, 331, 532, 811, 1206, 1729, 2432, 3331, 4494, 5942, 7760, 9975, 12692, 15944, 19858, 24473, 29941, 36308, 43752, 52327, 62239, 73551, 86499, 101155, 117788, 136479, 157532, 181038, 207338, 236534, 269005, 304865, 344534

Offset: 0

Colin Barker, Mar 23 2015

Also the number of partitions of 3*(n-2) into at most 6 parts. - Colin Barker, Apr 01 2015

			For n=3 the 3 partitions of 3*3 = 9 are [1,1,1,1,1,4], [1,1,1,1,2,3] and [1,1,1,2,2,2].

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,2,1,-4,3,3,-4,1,2,-4,1,2,-1).

Cf. A077043, A256313, A256314.

CoefficientList[Series[x^2*(x^8 + x^7 + 4*x^6 + 5*x^5 + 5*x^4 + 5*x^3 + 4*x^2 + x + 1)/((x - 1)^6*(x + 1)^3*(x^2 + 1)*(x^4 + x^3 + x^2 + x + 1)), {x, 0, 50}], x] (* Wesley Ivan Hurt, Feb 22 2017 *)
Table[Length@ IntegerPartitions[3 n, {6}], {n, 0, 40}] (* Michael De Vlieger, Feb 22 2017 *)

concat(0, vector(40, n, k=0; forpart(p=3*n, k++, , [6,6]); k))

concat([0,0], Vec(x^2*(x^8+x^7+4*x^6+5*x^5+5*x^4+5*x^3+4*x^2+x+1) / ((x-1)^6*(x+1)^3*(x^2+1)*(x^4+x^3+x^2+x+1)) + O(x^100)))

G.f.: x^2*(x^8+x^7+4*x^6+5*x^5+5*x^4+5*x^3+4*x^2+x+1) / ((x-1)^6*(x+1)^3*(x^2+1)*(x^4+x^3+x^2+x+1)).

A238410 a(n) = floor((3(n-1)^2 + 1)/2).

Original entry on oeis.org

0, 2, 6, 14, 24, 38, 54, 74, 96, 122, 150, 182, 216, 254, 294, 338, 384, 434, 486, 542, 600, 662, 726, 794, 864, 938, 1014, 1094, 1176, 1262, 1350, 1442, 1536, 1634, 1734, 1838, 1944, 2054, 2166, 2282, 2400, 2522, 2646, 2774, 2904, 3038, 3174, 3314, 3456, 3602, 3750, 3902, 4056, 4214, 4374, 4538, 4704

Offset: 1

Emeric Deutsch, Feb 27 2014

a(n) = the eccentric connectivity index of the path P[n] on n vertices. The eccentric connectivity index of a simple connected graph G is defined to be the sum over all vertices i of G of the product E(i)D(i), where E(i) is the eccentricity and D(i) is the degree of vertex i. For example, a(4)=14 because the vertices of P[4] have degrees 1,2,2,1 and eccentricities 3,2,2,3; we have 1*3 + 2*2 + 2*2 + 1*3 = 14.
From Paul Curtz, Feb 23 2023: (Start)
East spoke of the hexagonal spiral using A004526 with a single 0:
.
            43  42  42  41  41  40
          43  28  28  27  27  26  40
        44  29  17  16  16  15  26  39
      44  29  17   8   8   7  15  25  39
    45  30  18   9   3   2   7  14  25  38
  45  30  18   9   3   0---2---6--14--24--38-->
    31  19  10   4   1   1   6  13  24  37
      31  19  10   4   5   5  13  23  37
        32  20  11  11  12  12  23  36
          32  20  21  21  22  22  36
            33  33  34  34  35  35
.
Other spokes are A032528, A005449, A227017, A045943, A005448, A212959, A000326, A282513, A143689, and A104249. (End)

Matthew House, Table of n, a(n) for n = 1..10000
M. J. Morgan, S. Mukwembi and H. C. Swart, On the eccentric connectivity index of a graph, Discrete Math., 311, 2011, 1229-1234.
B. Zhou and Zh. Du, On eccentric connectivity index, Comm. Math. Comp. Chem. (MATCH), 63, 2010, 181-198.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A004526, A077043.

(Cf. A056105, A056107.)

a := proc (n) options operator, arrow: floor((3/2)*(n-1)^2+1/2) end proc: seq(a(n), n = 1 .. 70);

Table[Floor[(3(n-1)^2+1)/2],{n,80}]  (* or *) LinearRecurrence[{2,0,-2,1},{0,2,6,14},80] (* Harvey P. Dale, Apr 30 2022 *)

a(n)=(3*(n-1)^2 + 1)\2 \\ Charles R Greathouse IV, Feb 15 2017

a(n) = (3*n)^2/6 for n even and a(n) = ((3*n)^2 + 3)/6 for n odd. - Miquel Cerda, Jun 17 2016
From Ilya Gutkovskiy, Jun 17 2016: (Start)
G.f.: 2*x^2*(1 + x + x^2)/((1 - x)^3*(1 + x)).
a(n) = (6*n^2 - 12*n + 7 + (-1)^n)/4.
a(n) = 2* A077043(n-1). (End)
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). - Matthew House, Feb 15 2017
Sum_{n>=2} 1/a(n) = Pi^2/36 + tanh(Pi/(2*sqrt(3)))*Pi/(2*sqrt(3)). - Amiram Eldar, Mar 12 2023

A256226 Number of partitions of 6n into 6 parts.

Original entry on oeis.org

0, 1, 11, 58, 199, 532, 1206, 2432, 4494, 7760, 12692, 19858, 29941, 43752, 62239, 86499, 117788, 157532, 207338, 269005, 344534, 436140, 546261, 677571, 832989, 1015691, 1229120, 1476997, 1763332, 2092435, 2468926, 2897747, 3384171, 3933815, 4552649, 5247008

Offset: 0

Colin Barker, Mar 19 2015

			For n=2, the 11 partitions of 12 are Xs = [7,1,1,1,1,1], [6,2,1,1,1,1], [5,3,1,1,1,1], [4,4,1,1,1,1], [5,2,2,1,1,1], [4,3,2,1,1,1], [3,3,3,1,1,1], [4,2,2,2,1,1], [3,3,2,2,1,1], [3,2,2,2,2,1] and [2,2,2,2,2,2].

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-5,0,5,-3,-3,5,0,-5,4,-1).

Cf. A001402, A077043, A238340, A256225.

CoefficientList[Series[x (3 x^7 + 14 x^6 + 21 x^5 + 21 x^4 + 22 x^3 + 19 x^2 + 7 x + 1) / ((x - 1)^6 (x + 1) (x^4 + x^3 + x^2 + x + 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 20 2015 *)

concat(0, Vec(x*(3*x^7+14*x^6+21*x^5+21*x^4+22*x^3+19*x^2+7*x+1)/((x-1)^6*(x+1)*(x^4+x^3+x^2+x+1)) + O(x^100)))

concat(0, vector(35, n, k=0; forpart(p=6*n, k++, , [6,6]); k)) \\ Colin Barker, Mar 21 2015

G.f.: x*(3*x^7+14*x^6+21*x^5+21*x^4+22*x^3+19*x^2+7*x+1) / ((x-1)^6*(x+1)*(x^4+x^3+x^2+x+1)).

A330707 a(n) = ( 3*n^2 + n - 1 + (-1)^floor(n/2) )/4.

Original entry on oeis.org

0, 1, 3, 7, 13, 20, 28, 38, 50, 63, 77, 93, 111, 130, 150, 172, 196, 221, 247, 275, 305, 336, 368, 402, 438, 475, 513, 553, 595, 638, 682, 728, 776, 825, 875, 927, 981, 1036, 1092, 1150, 1210, 1271, 1333, 1397, 1463

Offset: 0

Paul Curtz, Dec 27 2019

Essentially four odds followed by four evens.
Last digit is neither 4 nor 9.
Essentially twice or twin sequences in the hexagonal spiral from A002265.
                  21  21  21  22  22  22  22
                21  14  14  14  14  15  15  23
              20  13   8   8   8   9   9  15  23
            20  13   8   4   4   4   4   9  15  23
          20  13   7   3   1   1   1   5   9  16  23
        20  13   7   3   1   0   0   2   5  10  16  24
          19  12   7   3   0   0   2   5  10  16  24
            19  12   7   3   2   2   5  10  16  24
              19  12   6   6   6   6  10  17  24
                19  12  11  11  11  11  17  25
                  18  18  18  18  17  17  25
.
There are 12 twin sequences. 6 of them (A001859, A006578, A077043, A231559, A024219, A281026) are in the OEIS. a(n) is the seventh.
  0, 1, 3, 7, 13, 20, 28, 38, 50, ...
  1, 2, 4, 6,  7,  8, 10, 12, 13, ...
  1, 2, 2, 1,  1,  2,  2,  1,  1, ... period 4. See A014695.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-4,4,-3,1).

Cf. A001859, A002265, A006578, A014695, A016921, A024219, A033579, A077043, A231559, A281026.

[(3*n^2+n-1+ (-1)^Floor(n/2))/4: n in [0..60]]; // G. C. Greubel, Dec 30 2019

seq((3*n^2+n-1+sqrt(2)*sin((2*n+1)*Pi/4))/4, n = 0..60); # G. C. Greubel, Dec 30 2019

LinearRecurrence[{3,-4,4,-3,1}, {0,1,3,7,13}, 60] (* Amiram Eldar, Dec 27 2019 *)

concat(0, Vec(x*(1 + 2*x^2) / ((1 - x)^3*(1 + x^2)) + O(x^60))) \\ Colin Barker, Dec 27 2019

[(3*n^2+n-1+(-1)^floor(n/2))/4 for n in (0..60)] # G. C. Greubel, Dec 30 2019

a(n) = A231559(-n).
a(1+2*n) + a(2+2*n) = A033579(n+1).
a(40+n) - a(n) = 1210, 1270, 1330, 1390, 1450, ... . See 10*A016921(n).
From Colin Barker, Dec 27 2019: (Start)
G.f.: x*(1 + 2*x^2) / ((1 - x)^3*(1 + x^2)).
a(n) = 3*a(n-1) - 4*a(n-2) + 4*a(n-3) - 3*a(n-4) + a(n-5) for n>4.
(End)
E.g.f.: (cos(x) + sin(x) + (-1 + 4*x + 3*x^2)*exp(x))/4. - Stefano Spezia, Dec 27 2019
a(n) = ( 3*n^2 + n - 1 + sqrt(2)*sin((2*n+1)*Pi/4) )/4 = ( 3*n^2 + n - 1 + (-1)^floor(n/2) )/4. - G. C. Greubel, Dec 30 2019

A227568 Largest k such that a partition of n into distinct parts with boundary size k exists.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10

Offset: 0

Alois P. Heinz, Jul 16 2013

nonn

The boundary size is the number of parts having fewer than two neighbors.

Alois P. Heinz, Table of n, a(n) for n = 0..2000

Where records occur: A077043.

Cf. A227345, A227551.

b:= proc(n, i, t) option remember; `if`(n=0, `if`(t>1, 1, 0),
      `if`(i<1, 0, max(`if`(t>1, 1, 0)+b(n, i-1, iquo(t, 2)),
      `if`(i>n, 0, `if`(t=2, 1, 0)+b(n-i, i-1, iquo(t, 2)+2)))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..100);

b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[t > 1, 1, 0], If[i < 1, 0, Max[If[t > 1, 1, 0] + b[n, i - 1, Quotient[t, 2]], If[i > n, 0, If[t == 2, 1, 0] + b[n - i, i - 1, Quotient[t, 2] + 2]]]]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, May 21 2018, translated from Maple *)

a(n) = max { k : A227345(n,k) > 0 } = max { k : A227551(n,k) > 0 }.

a(n) = floor(2*sqrt(n/3)).

A256313 Number of partitions of 3n into exactly 4 parts.

Original entry on oeis.org

0, 0, 2, 6, 15, 27, 47, 72, 108, 150, 206, 270, 351, 441, 551, 672, 816, 972, 1154, 1350, 1575, 1815, 2087, 2376, 2700, 3042, 3422, 3822, 4263, 4725, 5231, 5760, 6336, 6936, 7586, 8262, 8991, 9747, 10559, 11400, 12300, 13230, 14222, 15246, 16335, 17457

Offset: 0

Colin Barker, Mar 23 2015

			For n=3 the 6 partitions of 3*3 = 9 are [1,1,1,6], [1,1,2,5], [1,1,3,4], [1,2,2,4], [1,2,3,3] and [2,2,2,3].

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,2,-2,0,2,-1).

Cf. A077043, A256314, A256315.

LinearRecurrence[{2,0,-2,2,-2,0,2,-1},{0,0,2,6,15,27,47,72},60] (* Harvey P. Dale, Jul 18 2021 *)

concat(0, vector(40, n, k=0; forpart(p=3*n, k++, , [4,4]); k))

concat([0,0], Vec(x^2*(x^2+2)*(x^2+x+1)/((x-1)^4*(x+1)^2*(x^2+1)) + O(x^100)))

G.f.: x^2*(x^2+2)*(x^2+x+1) / ((x-1)^4*(x+1)^2*(x^2+1)).

a(n) = (6*n^3+6*n^2-3*n-5+(3*n+1)*(-1)^n+2*((-1)^((2*n-1+(-1)^n)/4)+(-1)^((2*n+1-(-1)^n)/4)))/32. - Luce ETIENNE, Feb 17 2017

A256314 Number of partitions of 3n into exactly 5 parts.

Original entry on oeis.org

0, 0, 1, 5, 13, 30, 57, 101, 164, 255, 377, 540, 748, 1014, 1342, 1747, 2233, 2818, 3507, 4319, 5260, 6351, 7599, 9027, 10642, 12470, 14518, 16814, 19366, 22204, 25337, 28796, 32591, 36756, 41301, 46262, 51649, 57501, 63829, 70673, 78045, 85987, 94512

Offset: 0

Colin Barker, Mar 23 2015

			For n=3 the 5 partitions of 3*3 = 9 are [1,1,1,1,5], [1,1,1,2,4], [1,1,1,3,3], [1,1,2,2,3] and [1,2,2,2,2].

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,2,-1,-2,2,1,-2,2,0,-2,1).

Cf. A077043, A256313, A256315.

Table[Length[IntegerPartitions[3n,{5}]],{n,0,50}] (* Harvey P. Dale, Jul 21 2019 *)

concat(0, vector(40, n, k=0; forpart(p=3*n, k++, , [5,5]); k))

concat([0,0], Vec(-x^2*(2*x^7+3*x^6+4*x^5+5*x^4+6*x^3+3*x^2+3*x+1) / ((x-1)^5*(x+1)^2*(x^2+1)*(x^4+x^3+x^2+x+1)) + O(x^100)))

G.f.: -x^2*(2*x^7+3*x^6+4*x^5+5*x^4+6*x^3+3*x^2+3*x+1) / ((x-1)^5*(x+1)^2*(x^2+1)*(x^4+x^3+x^2+x+1)).

A331952 a(n) = (-7 + (-1)^(1+n) + 6*n^2) / 8.

Original entry on oeis.org

-1, 0, 2, 6, 11, 18, 26, 36, 47, 60, 74, 90, 107, 126, 146, 168, 191, 216, 242, 270, 299, 330, 362, 396, 431, 468, 506, 546, 587, 630, 674, 720, 767, 816, 866, 918, 971, 1026, 1082, 1140, 1199, 1260, 1322, 1386, 1451, 1518, 1586, 1656, 1727, 1800, 1874, 1950, 2027

Offset: 0

Paul Curtz, Feb 02 2020

a(n+1) is once in the hexagonal spiral in A330707. a(n+2) is twice in the same spiral.
a(n) has one odd followed by three evens.
Difference table:
  -1, 0, 2, 6, 11, 18, 26, 36, ... = a(n)
   1, 2, 4, 5,  7,  8, 10, 11, ... = A001651(n+1)
   1, 2, 1, 2,  1,  2,  1,  2, ... = A000034.

			G.f. = -1 + 2*x^2 + 6*x^3 + 11*x^4 + 18*x^5 + 26*x^6 + 36*x^7 + 47*x^8 + ... - _Michael Somos_, Sep 08 2023

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).
Index entries for two-way infinite sequences

Cf. A000034, A001651, A158463, 6*A014105, 2*A003154, A152746(n+1), A008585(1+n).

Equals 2 less than A084684, 1 less than A077043, and 1 more than A276382(n-1). - Greg Dresden, Feb 22 2020

a:=[-1,0,2,6]; [n le 4 select a[n] else 2*Self(n-1)-2*Self(n-3)+Self(n-4): n in [1..45]]; // Marius A. Burtea, Feb 02 2020

LinearRecurrence[{2, 0, -2, 1}, {-1, 0, 2, 6}, 100] (* Amiram Eldar, Feb 02 2020 *)
a[n_] := Floor[(n^2 - 1)*3/4]; (* Michael Somos, Sep 08 2023 *)

Vec(-(1 - 2*x - 2*x^2) / ((1 - x)^3*(1 + x)) + O(x^40)) \\ Colin Barker, Feb 03 2020

{a(n) = (n^2 - 1)*3\4}; /* Michael Somos, Sep 08 2023 */

a(-n) = a(n).
a(20+n) - a(n) = 30*(10+n).
a(2+n) = a(n) + 3*(1+n), a(0)=-1 and a(1)=0.
a(4*n) = 12*n^2 - 1, a(1+4*n) = 6*n*(1+2*n), a(2+4*n) = 2 + 12*n*(1+n), a(3+4*n) = 6*(1+n)*(1+2*n) for n>= 0.
From Colin Barker, Feb 02 2020: (Start)
G.f.: -(1 - 2*x - 2*x^2) / ((1 - x)^3*(1 + x)).
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n>3.
a(n) = (-7 + (-1)^(1+n) + 6*n^2) / 8.
(End)
E.g.f.: (1/8)*(exp(x)*(6*x^2 + 6*x - 7) - exp(-x)). - Stefano Spezia, Feb 02 2020 after Colin Barker
a(n) = floor((n^2 - 1)*3/4). - Michael Somos, Sep 09 2023

a(42)-a(52) from Stefano Spezia, Feb 02 2020

A133458 The size of the largest antichain in the 7-dimensional hypercubic lattice of size n; also the coefficient of x^floor(7*(n-1)/2) in (1 + x + ... + x^(n-1))^7.

Original entry on oeis.org

1, 35, 393, 2128, 8135, 24017, 60691, 134512, 273127, 512365, 908755, 1528688, 2473325, 3852919, 5832765, 8582336, 12354469, 17395119, 24072133, 32726960, 43874139, 57971221, 75715487, 97702640, 124853275, 157924585, 198105727

Offset: 1

Leonid Chindelevitch (leonidus(AT)mit.edu), Dec 22 2007

nonn

The middle coefficients for dimension d>=1 are in A000012, A000027, A077043, A005900, A077044, A071816, here, the d-th row in A077042.

For d=8 the sequence starts 1, 70, 1107, 8092, 38165, 135954, 398567, 1012664, 2306025, ... and for d=9 it starts 1, 126, 3139, 30276, 180325, 767394, 2636263, 7635987, 19610233, ... - R. J. Mathar, Sep 04 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
R. P. Stanley, Weyl groups, the Hard Lefschetz Theorem and the Sperner property, SIAM J. Alg. Disc. Meth. 1 (2) (1980) 168, see eq. (4).

Cf. A077043, A005900, A077044, A071816.

[-25*(-1)^n/512 +2261*n^2/23040 +25/512 +5887*n^6/11520 -77*(-1)^n*n^4/1536 +847*n^4/4608 -91*(-1)^n*n^2/1536 : n in [1..40]]; // Vincenzo Librandi, Sep 07 2011

f:=(L,d)->(sum(x^k,k=0..L-1))^d; A:=[seq(coeff(f(j,7),x,floor(7*(j-1)/2)),j=1..25)];
A133458 := proc(n) -25/512*(-1)^n +2261/23040*n^2 -91/1536*(-1)^n*n^2 -77/1536*(-1)^n*n^4 +847/4608*n^4 +5887/11520*n^6 +25/512 ; end proc: # R. J. Mathar, Sep 05 2011

From R. J. Mathar, Feb 19 2010: (Start)
a(n)= 2*a(n-1) +4*a(n-2) -10*a(n-3) -5*a(n-4) +20*a(n-5) -20*a(n-7) +5*a(n-8) +10*a(n-9) -4*a(n-10) -2*a(n-11) +a(n-12).
G.f.: x*(1+33*x +319*x^2 +1212*x^3 +2662*x^4 +3320*x^5 +2662*x^6 +1212*x^7 +319*x^8 +33*x^9 +x^10)/ ((1+x)^5 * (1-x)^7).
a(n) = -25*(-1)^n/512 +2261*n^2/23040 +25/512 +5887*n^6/11520 -77*(-1)^n*n^4/1536 +847*n^4/4608 -91*(-1)^n*n^2/1536. (End)

More terms from R. J. Mathar, Feb 19 2010

A212503 Number of (w,x,y,z) with all terms in {1,...,n} and w<2x and y<2z.

Original entry on oeis.org

0, 1, 9, 49, 144, 361, 729, 1369, 2304, 3721, 5625, 8281, 11664, 16129, 21609, 28561, 36864, 47089, 59049, 73441, 90000, 109561, 131769, 157609, 186624, 219961, 257049, 299209, 345744, 398161, 455625, 519841, 589824, 667489, 751689

Offset: 0

Clark Kimberling, May 19 2012

For a guide to related sequences, see A211795.

Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).

Cf. A211795.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w < 2 x && y < 2 z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]]   (* A212503 *)
LinearRecurrence[{2,2,-6,0,6,-2,-2,1},{0,1,9,49,144,361,729,1369},40] (* Harvey P. Dale, Jun 14 2017 *)

a(n) = (A077043(n))^2.
a(n) = 2*a(n-1)+2*a(n-2)-6*a(n-3)+6*a(n-5)-2*a(n-6)-2*a(n-7)+a(n-8).
G.f.: x*(1+x+x^2)*(1+6*x+22*x^2+6*x^3+x^4)/((1+x)^3*(1-x)^5). [Bruno Berselli, May 31 2012]

cp's OEIS Frontend

A256315 Number of partitions of 3n into exactly 6 parts.

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A238410 a(n) = floor((3(n-1)^2 + 1)/2).

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Maple

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A256226 Number of partitions of 6n into 6 parts.

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A330707 a(n) = ( 3*n^2 + n - 1 + (-1)^floor(n/2) )/4.

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A227568 Largest k such that a partition of n into distinct parts with boundary size k exists.

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A256313 Number of partitions of 3n into exactly 4 parts.

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A256314 Number of partitions of 3n into exactly 5 parts.

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