A026810 -id:A026810 - cp's OEIS frontend

A319854 Number of ways to write n as the sum of 4 positive integers a, b, c, d such that d < b and a/b - c/d = (a - c)/(b + d).

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

Offset: 1

Hugo Pfoertner and Rainer Rosenthal, Sep 29 2018

Number of ways to write n as the sum of 4 positive integers a, b, c, d such that d < b and a*d^2 = b^2*c. - Robert Israel, Oct 04 2018

			a(8) = 1: 4/2 - 1/1 = (4 - 1)/(2 + 1) = 1;
a(11) = 1: 4/4 - 1/2 = (4 - 1)/(4 + 2) = 1/2;
a(13) = 1: 8/2 - 2/1 = (8 - 2)/(2 + 1) = 2;
a(14) = 2: 4/6 - 1/3 = (4 - 1)/(6 + 3) = 1/3, 9/3 - 1/1 = (9 - 1)/(3 + 1) = 2;
a(16) = 1: 8/4 - 2/2 = (8 - 2)/(4 + 2) = 1;
a(17) = 1: 4/8 - 1/4 = (4 - 1)/(8 + 4) = 1/4;
a(18) = 3: 9/3 - 4/2 = (9 - 4)/(3 + 2) = 1, 9/6 - 1/2 = (9 - 1)/(6 + 2) = 1, 12/2 - 3/1 = (12 - 3)/(2 + 1) = 3.

Robert Israel, Table of n, a(n) for n = 1..10000 (first 500 terms from Hugo Pfoertner)

Cf. A026810, A319853, A320311, A320312, A320313.

N:= 1000: # for a(1)..a(N)
V:= Vector(N):
for d from 1 to N/2 do
  for b from d+1 to N-d do
    u:= d^2/igcd(b,d)^2;
    for c from u by u  do
      v:= c*b^2/d^2+b+c+d;
      if v > N then break fi;
      V[v]:= V[v]+1
od od od:
convert(V,list); # Robert Israel, Oct 04 2018

M = 100; Clear[V]; V[_] = 0;
For[d = 1, d <= M/2, d++,
  For[b = d+1, b <= M-d, b++,
    u = d^2/GCD[b, d]^2;
    For[c = u, True, c = c+u,
      v = c*b^2/d^2 + b + c + d;
      If[v > M, Break[]];
      V[v] = V[v]+1
]]];
Array[V, M] (* Jean-François Alcover, Apr 02 2019, after Robert Israel *)

m=84;v=vector(m);for(a=1,m,for(b=1,m,for(c=1,m,for(d=1,b-1,n=a+b+c+d;if(n<=m,if((a/b-c/d)==((a-c)/(b+d)),v[n]++))))));v

A320312 Number of ways to write n as the sum of 4 positive integers a, b, c, d such that b != d and a/b - c/d = (a - c)/(b - d).

Original entry on oeis.org

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

Offset: 1

Hugo Pfoertner, Oct 10 2018

nonn

Hugo Pfoertner, Table of n, a(n) for n = 1..1000

Cf. A026810, A319853, A319854, A320311, A320313.

m=80;v=vector(m);for(a=1,m,for(b=1,m,for(c=1,m,for(d=1,m,n=a+b+c+d;if(n<=m,if(b!=d&&a/b-c/d==(a-c)/(b-d),v[n]++))))));v

A320313 Number of ways to write n as the sum of 4 positive integers a, b, c, d such that d < b and a/b - c/d = (a - c)/(b - d).

Original entry on oeis.org

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

Offset: 1

Hugo Pfoertner, Oct 10 2018

nonn

Hugo Pfoertner, Table of n, a(n) for n = 1..1000

Cf. A026810, A319853, A319854, A320311, A320312.

m=86;v=vector(m);for(a=1,m,for(b=1,m,for(c=1,m,for(d=1,b-1,n=a+b+c+d;if(n<=m,if(a/b-c/d==(a-c)/(b-d),v[n]++))))));v

A308733 Sum of the smallest parts of the partitions of n into 4 parts.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 3, 6, 7, 11, 14, 21, 25, 34, 41, 55, 64, 81, 95, 119, 136, 165, 189, 227, 256, 301, 339, 396, 441, 507, 564, 645, 711, 804, 885, 996, 1089, 1215, 1326, 1474, 1600, 1766, 1914, 2106, 2272, 2486, 2678, 2922, 3136, 3406, 3650, 3955, 4225, 4560

Offset: 0

Wesley Ivan Hurt, Jun 22 2019

nonn

			Figure 1: The partitions of n into 4 parts for n = 8, 9, ..
                                                         1+1+1+9
                                                         1+1+2+8
                                                         1+1+3+7
                                                         1+1+4+6
                                             1+1+1+8     1+1+5+5
                                             1+1+2+7     1+2+2+7
                                 1+1+1+7     1+1+3+6     1+2+3+6
                                 1+1+2+6     1+1+4+5     1+2+4+5
                                 1+1+3+5     1+2+2+6     1+3+3+5
                     1+1+1+6     1+1+4+4     1+2+3+5     1+3+4+4
         1+1+1+5     1+1+2+5     1+2+2+5     1+2+4+4     2+2+2+6
         1+1+2+4     1+1+3+4     1+2+3+4     1+3+3+4     2+2+3+5
         1+1+3+3     1+2+2+4     1+3+3+3     2+2+2+5     2+2+4+4
         1+2+2+3     1+2+3+3     2+2+2+4     2+2+3+4     2+3+3+4
         2+2+2+2     2+2+2+3     2+2+3+3     2+3+3+3     3+3+3+3
--------------------------------------------------------------------------
  n  |      8           9          10          11          12        ...
--------------------------------------------------------------------------
a(n) |      6           7          11          14          21        ...
--------------------------------------------------------------------------
- _Wesley Ivan Hurt_, Sep 07 2019

Index entries for sequences related to partitions

Cf. A026810, A308758, A308759, A308760, A308775.

Table[Sum[Sum[Sum[k, {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 100}]

a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} k.
a(n) = A308775(n) - A308758(n) - A308759(n) - A308760(n).
Conjectures from Colin Barker, Jun 23 2019: (Start)
G.f.: x^4 / ((1 - x)^5*(1 + x)^3*(1 + x^2)^2*(1 + x + x^2)).
a(n) = a(n-1) + a(n-2) + a(n-4) - 3*a(n-5) - a(n-6) + a(n-8) + 3*a(n-9) - a(n-10) - a(n-12) - a(n-13) + a(n-14) for n>13.
(End)

A308758 Sum of the third largest parts of the partitions of n into 4 parts.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 4, 7, 9, 15, 20, 29, 38, 51, 64, 86, 104, 131, 160, 198, 233, 284, 332, 396, 459, 538, 616, 719, 814, 934, 1056, 1203, 1344, 1521, 1692, 1899, 2103, 2343, 2580, 2866, 3139, 3461, 3784, 4156, 4518, 4944, 5360, 5840, 6314, 6852, 7384, 7997

Offset: 0

Wesley Ivan Hurt, Jun 22 2019

nonn

			Figure 1: The partitions of n into 4 parts for n = 8, 9, ..
                                                         1+1+1+9
                                                         1+1+2+8
                                                         1+1+3+7
                                                         1+1+4+6
                                             1+1+1+8     1+1+5+5
                                             1+1+2+7     1+2+2+7
                                 1+1+1+7     1+1+3+6     1+2+3+6
                                 1+1+2+6     1+1+4+5     1+2+4+5
                                 1+1+3+5     1+2+2+6     1+3+3+5
                     1+1+1+6     1+1+4+4     1+2+3+5     1+3+4+4
         1+1+1+5     1+1+2+5     1+2+2+5     1+2+4+4     2+2+2+6
         1+1+2+4     1+1+3+4     1+2+3+4     1+3+3+4     2+2+3+5
         1+1+3+3     1+2+2+4     1+3+3+3     2+2+2+5     2+2+4+4
         1+2+2+3     1+2+3+3     2+2+2+4     2+2+3+4     2+3+3+4
         2+2+2+2     2+2+2+3     2+2+3+3     2+3+3+3     3+3+3+3
--------------------------------------------------------------------------
  n  |      8           9          10          11          12        ...
--------------------------------------------------------------------------
a(n) |      7           9          15          20          29        ...
--------------------------------------------------------------------------
- _Wesley Ivan Hurt_, Sep 07 2019

Index entries for sequences related to partitions

Cf. A026810, A308733, A308759, A308760, A308775.

Table[Sum[Sum[Sum[j, {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 100}]
Table[Total[IntegerPartitions[n,{4}][[All,3]]],{n,0,60}] (* Harvey P. Dale, Dec 10 2021 *)

a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} j.
a(n) = A308775(n) - A308733(n) - A308759(n) - A308760(n).
Conjectures from Colin Barker, Jun 23 2019: (Start)
G.f.: x^4*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6) / ((1 - x)^5*(1 + x)^3*(1 + x^2)^2*(1 + x + x^2)^2).
a(n) = a(n-2) + 2*a(n-3) + 2*a(n-4) - 2*a(n-5) - 3*a(n-6) - 4*a(n-7) + 4*a(n-9) + 3*a(n-10) + 2*a(n-11) - 2*a(n-12) - 2*a(n-13) - a(n-14) + a(n-16) for n>15.
(End)

A308759 Sum of the second largest parts of the partitions of n into 4 parts.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 3, 5, 10, 13, 23, 30, 46, 59, 83, 103, 141, 170, 220, 265, 334, 392, 484, 563, 680, 784, 930, 1061, 1247, 1409, 1631, 1836, 2106, 2349, 2673, 2967, 3348, 3699, 4143, 4554, 5077, 5554, 6150, 6710, 7396, 8032, 8816, 9546, 10432, 11264, 12260

Offset: 0

Wesley Ivan Hurt, Jun 22 2019

nonn

			Figure 1: The partitions of n into 4 parts for n = 8, 9, ..
                                                         1+1+1+9
                                                         1+1+2+8
                                                         1+1+3+7
                                                         1+1+4+6
                                             1+1+1+8     1+1+5+5
                                             1+1+2+7     1+2+2+7
                                 1+1+1+7     1+1+3+6     1+2+3+6
                                 1+1+2+6     1+1+4+5     1+2+4+5
                                 1+1+3+5     1+2+2+6     1+3+3+5
                     1+1+1+6     1+1+4+4     1+2+3+5     1+3+4+4
         1+1+1+5     1+1+2+5     1+2+2+5     1+2+4+4     2+2+2+6
         1+1+2+4     1+1+3+4     1+2+3+4     1+3+3+4     2+2+3+5
         1+1+3+3     1+2+2+4     1+3+3+3     2+2+2+5     2+2+4+4
         1+2+2+3     1+2+3+3     2+2+2+4     2+2+3+4     2+3+3+4
         2+2+2+2     2+2+2+3     2+2+3+3     2+3+3+3     3+3+3+3
--------------------------------------------------------------------------
  n  |      8           9          10          11          12        ...
--------------------------------------------------------------------------
a(n) |     10          13          23          30          46        ...
--------------------------------------------------------------------------
- _Wesley Ivan Hurt_, Sep 07 2019

Index entries for sequences related to partitions

Cf. A026810, A308733, A308758, A308760, A308775.

Table[Total[IntegerPartitions[n,{4}][[All,2]]],{n,0,50}] (* Harvey P. Dale, Nov 08 2020 *)

a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} i.
a(n) = A308775(n) - A308733(n) - A308758(n) - A308760(n).
Conjectures from Colin Barker, Jun 23 2019: (Start)
G.f.: x^4*(1 + x + 2*x^2 + 2*x^3 + 3*x^4 + 2*x^5 + 2*x^6) / ((1 - x)^5*(1 + x)^3*(1 + x^2)^2*(1 + x + x^2)^2).
a(n) = a(n-2) + 2*a(n-3) + 2*a(n-4) - 2*a(n-5) - 3*a(n-6) - 4*a(n-7) + 4*a(n-9) + 3*a(n-10) + 2*a(n-11) - 2*a(n-12) - 2*a(n-13) - a(n-14) + a(n-16) for n>15.
(End)

A308760 Sum of the largest parts of the partitions of n into 4 parts.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 5, 9, 17, 25, 41, 57, 84, 112, 154, 197, 262, 325, 414, 506, 629, 751, 915, 1078, 1289, 1501, 1767, 2034, 2370, 2701, 3108, 3519, 4014, 4506, 5100, 5691, 6393, 7095, 7917, 8739, 9703, 10658, 11765, 12876, 14150, 15418, 16874, 18324, 19974

Offset: 0

Wesley Ivan Hurt, Jun 22 2019

nonn

			Figure 1: The partitions of n into 4 parts for n = 8, 9, ..
                                                         1+1+1+9
                                                         1+1+2+8
                                                         1+1+3+7
                                                         1+1+4+6
                                             1+1+1+8     1+1+5+5
                                             1+1+2+7     1+2+2+7
                                 1+1+1+7     1+1+3+6     1+2+3+6
                                 1+1+2+6     1+1+4+5     1+2+4+5
                                 1+1+3+5     1+2+2+6     1+3+3+5
                     1+1+1+6     1+1+4+4     1+2+3+5     1+3+4+4
         1+1+1+5     1+1+2+5     1+2+2+5     1+2+4+4     2+2+2+6
         1+1+2+4     1+1+3+4     1+2+3+4     1+3+3+4     2+2+3+5
         1+1+3+3     1+2+2+4     1+3+3+3     2+2+2+5     2+2+4+4
         1+2+2+3     1+2+3+3     2+2+2+4     2+2+3+4     2+3+3+4
         2+2+2+2     2+2+2+3     2+2+3+3     2+3+3+3     3+3+3+3
--------------------------------------------------------------------------
  n  |      8           9          10          11          12        ...
--------------------------------------------------------------------------
a(n) |     17          25          41          57          84        ...
--------------------------------------------------------------------------
- _Wesley Ivan Hurt_, Sep 07 2019

Index entries for sequences related to partitions

Cf. A001318, A026810, A308265, A308733, A308758, A308759, A308775.

Table[Sum[Sum[Sum[n - i - j - k, {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 100}]

a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} (n-i-j-k).
a(n) = A308775(n) - A308733(n) - A308758(n) - A308759(n).
Conjectures from Colin Barker, Jun 23 2019: (Start)
G.f.: x^4*(1 + 2*x + 4*x^2 + 5*x^3 + 6*x^4 + 4*x^5 + 3*x^6) / ((1 - x)^5*(1 + x)^3*(1 + x^2)^2*(1 + x + x^2)^2).
a(n) = a(n-2) + 2*a(n-3) + 2*a(n-4) - 2*a(n-5) - 3*a(n-6) - 4*a(n-7) + 4*a(n-9) + 3*a(n-10) + 2*a(n-11) - 2*a(n-12) - 2*a(n-13) - a(n-14) + a(n-16) for n>15.
(End)

A308775 Sum of all the parts in the partitions of n into 4 parts.

Original entry on oeis.org

0, 0, 0, 0, 4, 5, 12, 21, 40, 54, 90, 121, 180, 234, 322, 405, 544, 663, 846, 1026, 1280, 1512, 1848, 2162, 2592, 3000, 3536, 4050, 4732, 5365, 6180, 6975, 7968, 8910, 10098, 11235, 12636, 13986, 15618, 17199, 19120, 20951, 23142, 25284, 27808, 30240, 33120

Offset: 0

Wesley Ivan Hurt, Jun 23 2019

nonn

			Figure 1: The partitions of n into 4 parts for n = 8, 9, ..
                                                         1+1+1+9
                                                         1+1+2+8
                                                         1+1+3+7
                                                         1+1+4+6
                                             1+1+1+8     1+1+5+5
                                             1+1+2+7     1+2+2+7
                                 1+1+1+7     1+1+3+6     1+2+3+6
                                 1+1+2+6     1+1+4+5     1+2+4+5
                                 1+1+3+5     1+2+2+6     1+3+3+5
                     1+1+1+6     1+1+4+4     1+2+3+5     1+3+4+4
         1+1+1+5     1+1+2+5     1+2+2+5     1+2+4+4     2+2+2+6
         1+1+2+4     1+1+3+4     1+2+3+4     1+3+3+4     2+2+3+5
         1+1+3+3     1+2+2+4     1+3+3+3     2+2+2+5     2+2+4+4
         1+2+2+3     1+2+3+3     2+2+2+4     2+2+3+4     2+3+3+4
         2+2+2+2     2+2+2+3     2+2+3+3     2+3+3+3     3+3+3+3
--------------------------------------------------------------------------
  n  |      8           9          10          11          12        ...
--------------------------------------------------------------------------
a(n) |     40          54          90         121         180        ...
--------------------------------------------------------------------------
- _Wesley Ivan Hurt_, Sep 07 2019

Index entries for sequences related to partitions

Cf. A026810, A308733, A308758, A308759, A308760.

Table[n*Sum[Sum[Sum[1, {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 100}]

a(n) = n * A026810(n).
a(n) = A308733(n) + A308758(n) + A308759(n) + A308760(n).
Conjectures from Colin Barker, Jun 24 2019: (Start)
G.f.: x^4*(4 + 5*x + 8*x^2 + 8*x^3 + 10*x^4 + 7*x^5 + 6*x^6) / ((1 - x)^5*(1 + x)^3*(1 + x^2)^2*(1 + x + x^2)^2).
a(n) = a(n-2) + 2*a(n-3) + 2*a(n-4) - 2*a(n-5) - 3*a(n-6) - 4*a(n-7) + 4*a(n-9) + 3*a(n-10) + 2*a(n-11) - 2*a(n-12) - 2*a(n-13) - a(n-14) + a(n-16) for n>15.
(End)

A320311 Number of ways to write n as the sum of 4 positive integers a, b, c, d such that 1 - a/c = (1 - b/d)^2.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 3, 0, 5, 0, 6, 1, 6, 0, 8, 2, 9, 2, 10, 0, 15, 1, 14, 1, 15, 3, 15, 2, 17, 4, 19, 3, 21, 1, 21, 4, 26, 3, 25, 4, 24, 4, 27, 6, 29, 5, 31, 5, 30, 4, 36, 4, 37, 6, 34, 6, 41, 8, 36, 8, 43, 4, 42, 5, 44, 10, 44, 9, 45, 9, 45, 10, 48

Offset: 1

Hugo Pfoertner and Rainer Rosenthal Oct 10 2018

nonn

Cf. A026810, A319853, A319854, A320312, A320313.

m=74;v=vector(m);for(a=1,m,for(b=1,m,for(c=1,m,for(d=1,m,n=a+b+c+d;if(n<=m,if(1-a/c==(1-b/d)^2,v[n]++))))));v

A008641 Number of partitions of n into at most 12 parts.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 100, 133, 172, 224, 285, 366, 460, 582, 725, 905, 1116, 1380, 1686, 2063, 2503, 3036, 3655, 4401, 5262, 6290, 7476, 8877, 10489, 12384, 14552, 17084, 19978, 23334, 27156, 31570, 36578, 42333, 48849, 56297

Offset: 0

N. J. A. Sloane

With a different offset, number of partitions of n in which the greatest part is 12.

Also number of partitions of n into parts <= 12: a(n)=A026820(n,12). [Reinhard Zumkeller, Jan 21 2010]

A. Cayley, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 10, p. 415.
H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 2.

T. D. Noe, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 361
Index entries for linear recurrences with constant coefficients, signature (1, 1, 0, 0, -1, 0, -1, 0, 0, 0, 0, 1, -1, 0, 2, 1, 1, 0, 0, -1, -1, -2, -1, -1, 0, -2, 0, 1, 2, 2, 2, 2, 1, 1, 0, -1, -2, -1, -4, -1, -2, -1, 0, 1, 1, 2, 2, 2, 2, 1, 0, -2, 0, -1, -1, -2, -1, -1, 0, 0, 1, 1, 2, 0, -1, 1, 0, 0, 0, 0, -1, 0, -1, 0, 0, 1, 1, -1).

a(n) = A008284(n+12, 12), n >= 0.

Cf. A026810, A026811, A026812, A026813, A026814, A026815, A026816.

1/(1-x)/(1-x^2)/(1-x^3)/(1-x^4)/(1-x^5)/(1-x^6)/(1-x^7)/(1-x^8)/(1-x^9)/(1-x^10)/(1-x^11)/(1-x^12)
with(combstruct):ZL13:=[S,{S=Set(Cycle(Z,card<13))}, unlabeled]:seq(count(ZL13,size=n),n=0..46); # Zerinvary Lajos, Sep 24 2007
B:=[S,{S = Set(Sequence(Z,1 <= card),card <=12)},unlabelled]: seq(combstruct[count](B, size=n), n=0..46); # Zerinvary Lajos, Mar 21 2009

CoefficientList[ Series[ 1/ Product[ 1 - x^n, {n, 1, 12} ], {x, 0, 60} ], x ]
Table[ Length[ Select[ Partitions[n], First[ # ] == 12 & ]], {n, 1, 60} ]

G.f.: 1/Product_{k=1..12}(1-x^k).

a(n) = a(n-1) + a(n-2) - a(n-5) - a(n-7) + a(n-12) - a(n-13) + 2*a(n-15) + a(n-16) + a(n-17) - a(n-20) - a(n-21) - 2*a(n-22) - a(n-23) - a(n-24) - 2*a(n-26) + a(n-28) + 2*a(n-29) + 2*a(n-30) + 2*a(n-31) + 2*a(n-32) + a(n-33) + a(n-34) - a(n-36) - 2*a(n-37) - a(n-38) - 4*a(n-39) - a(n-40) - 2*a(n-41) - a(n-42) + a(n-44) + a(n-45) + 2*a(n-46) + 2a(n-47) + 2*a(n-48) + 2*a(n-49) + a(n-50) - 2*a(n-52) - a(n-54) - a(n-55) - 2*a(n-56) - a(n-57) - a(n-58) + a(n-61) + a(n-62) + 2*a(n-63) - a(n-65) + a(n-66) - a(n-71) - a(n-73) + a(n-76) + a(n-77) - a(n-78). - David Neil McGrath, Jul 28 2015

More terms from Robert G. Wilson v, Dec 11 2000

cp's OEIS Frontend

A319854 Number of ways to write n as the sum of 4 positive integers a, b, c, d such that d < b and a/b - c/d = (a - c)/(b + d).

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A320312 Number of ways to write n as the sum of 4 positive integers a, b, c, d such that b != d and a/b - c/d = (a - c)/(b - d).

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A320313 Number of ways to write n as the sum of 4 positive integers a, b, c, d such that d < b and a/b - c/d = (a - c)/(b - d).

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A308733 Sum of the smallest parts of the partitions of n into 4 parts.

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A308758 Sum of the third largest parts of the partitions of n into 4 parts.

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A308759 Sum of the second largest parts of the partitions of n into 4 parts.

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A308760 Sum of the largest parts of the partitions of n into 4 parts.

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A308775 Sum of all the parts in the partitions of n into 4 parts.

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A320311 Number of ways to write n as the sum of 4 positive integers a, b, c, d such that 1 - a/c = (1 - b/d)^2.