A069905 -id:A069905 - cp's OEIS frontend

A350042 Sum of all the parts in the partitions of n into 3 positive integer parts.

0, 0, 0, 3, 4, 10, 18, 28, 40, 63, 80, 110, 144, 182, 224, 285, 336, 408, 486, 570, 660, 777, 880, 1012, 1152, 1300, 1456, 1647, 1820, 2030, 2250, 2480, 2720, 3003, 3264, 3570, 3888, 4218, 4560, 4953, 5320, 5740, 6174, 6622, 7084, 7605, 8096, 8648, 9216, 9800, 10400

Offset: 0

Wesley Ivan Hurt, Dec 10 2021

nonn

			a(9) = 63 since we have the partitions (1,1,7), (1,2,6), (1,3,5), (1,4,4), (2,2,5), (2,3,4) and (3,3,3). Since the parts in each partition sum to 9 and we have 7 partitions, a(9) = 9*7 = 63.

Winston de Greef, Table of n, a(n) for n = 0..10000
Index entries for sequences related to partitions

Cf. A069905.

a(n) = floor((n^2+6)/12) * n \\ Winston de Greef, Oct 02 2023

a(n) = n * A069905(n).

A360240 Weakly decreasing triples of positive integers sorted lexicographically and concatenated.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Feb 11 2023

nonn

			Triples begin: (1,1,1), (2,1,1), (2,2,1), (2,2,2), (3,1,1), (3,2,1), (3,2,2), (3,3,1), (3,3,2), (3,3,3), ...

The triples have sums A070770.
Positions of first appearances are A158842.
For pairs instead of triples we have A330709 + 1.
The zero-based version is A331195.
- The first part is A360010 = A056556 + 1.
- The second part is A194848 = A056557 + 1.
- The third part is A333516 = A056558 + 1.
Cf. A002024, A003056, A069905.

nn=9;Join@@Select[Tuples[Range[nn],3],GreaterEqual@@#&]

from math import isqrt, comb
from sympy import integer_nthroot
def A360240(n): return (m:=integer_nthroot((n-1<<1)+6,3)[0])+(n>3*comb(m+2,3)) if (a:=n%3)==1 else (k:=isqrt(r:=(b:=(n-1)//3)+1-comb((m:=integer_nthroot((n-1<<1)-1,3)[0])-(b(k<<2)*(k+1)+1) if a==2 else 1+(r:=(b:=(n-1)//3)-comb((m:=integer_nthroot((n-1<<1)-3,3)[0])+(b>=comb(m+2,3))+1,3))-comb((k:=isqrt(m:=r+1<<1))+(m>k*(k+1)),2) # Chai Wah Wu, Jun 07 2025

a(n) = A331195(n-1) + 1.

A158138 Number of nondecreasing integer sequences of length 4 with sum zero and sum of absolute values 2n.

Original entry on oeis.org

1, 4, 6, 11, 13, 22, 24, 35, 39, 52, 56, 73, 77, 96, 102, 123, 129, 154, 160, 187, 195, 224, 232, 265, 273, 308, 318, 355, 365, 406, 416, 459, 471, 516, 528, 577, 589, 640, 654, 707, 721, 778, 792, 851, 867, 928, 944, 1009, 1025, 1092, 1110, 1179, 1197, 1270, 1288

Offset: 1

R. H. Hardin, Mar 13 2009

nonn

a(n) = A000041(n)^2 for n<=2

a(n) = A000041(n)^2 - cumulative A000712(2*n-1-length), 0 <= 2*n-1-length <= floor(n/2) [empirical].

			For n = 6, we count the possible concatenations of the 4 pairs in the list (-6,0),(-5,-1),(-4,-2),(-3,-3) with their negative reversed correspondants (starting with (-6,0,0,6)), giving (6/2 + 1)^2 = 16 quadruples, plus the 3 quadruples (-6,1,1,4), (-6,1,2,3), (-6,2,2,2) and their 3 negative reversed correspondants, giving a total of 22 possibilities. - _Georg Fischer_, Apr 20 2022

R. H. Hardin, Table of n, a(n) for n=1..999

Cf. A069905, A158139-A158184 (for length 5..50).

# empirical
function a(n) { s=1; for(i=1; i

a(n) = (floor(n/2) + 1)^2 + 2*A069905(n). - Georg Fischer, Apr 20 2022

A328863 Number of partitions of 2n that describe the degree sequence of exactly one labeled multigraph with no loops.

Original entry on oeis.org

1, 2, 4, 6, 9, 14, 19, 27, 37, 50, 66, 89, 115, 151, 195, 252, 321, 412, 520, 660, 829, 1042, 1299, 1623, 2010, 2492, 3071, 3783, 4635, 5679, 6922, 8434, 10234, 12406, 14985, 18085, 21751, 26135, 31312, 37471, 44723, 53321, 63415, 75336, 89303, 105734, 124938

Offset: 1

Peter Kagey, Oct 28 2019

nonn

Also the number of partitions of 2*n either with largest part equal to n or with three parts and largest part less than n.

			For n = 4, the a(4) = 6 partitions of 2*4 = 8 that describe a degree sequence of exactly one labeled multigraph are
  4 + 4,
  4 + 3 + 1,
  4 + 2 + 2,
  4 + 2 + 1 + 1,
  4 + 1 + 1 + 1 + 1, and
  3 + 3 + 2.

Peter Kagey, Table of n, a(n) for n = 1..10000

Cf. A000041, A069905, A209816.

a(n) = A000041(n) + A069905(n).

A338200 The number of similarity classes of pointed reflection spaces of residue two in an n-dimensional vector space over GF(2).

Original entry on oeis.org

0, 0, 1, 2, 4, 6, 9, 12, 17, 21, 27, 33, 41, 48, 58, 67, 79, 90, 104, 117, 134, 149, 168, 186, 208, 228, 253, 276, 304, 330, 361, 390, 425, 457, 495, 531, 573, 612, 658, 701, 751, 798, 852, 903, 962, 1017, 1080, 1140, 1208, 1272, 1345, 1414, 1492, 1566, 1649

Offset: 1

Masaya Tomie, Oct 16 2020

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Saeid Azam, Masaya Tomie, and Yoji Yoshii, Classification of pointed reflection spaces, Osaka J. Math. (2021) Vol. 58, 563-589.
Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-2,0,0,1,1,-1).

Cf. A069905.

F[n_] := If[EvenQ[n],
  n (n - 2)/8 +
   2*Sum[Length[IntegerPartitions[k, {3}]], {k, 3, n/2}] +
   Length[IntegerPartitions[(n + 2)/2, {3}]],
  2*Floor[(n - 1)/4]*Floor[(n + 1)/4] +
   2*Sum[Length[IntegerPartitions[k, {3}]], {k, 3, (n - 1)/2}] +
   Length[IntegerPartitions[(n + 1)/2, {3}]] +
   Length[IntegerPartitions[(n + 3)/2, {3}]]]
(* Second program: *)
LinearRecurrence[{1,1,0,0,-2,0,0,1,1,-1}, {0,0,1,2,4,6,9,12,17,21}, 55] (* Jean-François Alcover, Nov 13 2020 *)

concat([0,0], Vec((1 + x + x^2 - x^4 - x^5)/((1 - x)^4*(1 + x)^2*(1 + x^2)*(1 + x + x^2)) + O(x^50))) \\ Andrew Howroyd, Oct 29 2020

a(n) = (1/8)*n*(n-2) + 2*(Sum_{k=3..n/2} p(k,3)) + p((n+2)/2,3) if n is even; a(n) = 2*floor((n-1)/4)*floor((n+1)/4) + 2*(Sum_{k=3..(n-1)/2} p(k,3)) + p((n+1)/2,3) + p((n+3)/2,3) if n is odd, where p(k,3) = A069905(k) is the number of partitions of k into three parts.
From Andrew Howroyd, Oct 29 2020: (Start)
a(n) = a(n-1) + a(n-2) - 2*a(n-5) + a(n-8) + a(n-9) - a(n-10) for n > 10.
G.f.: x^3*(1 + x + x^2 - x^4 - x^5)/((1 - x)^4*(1 + x)^2*(1 + x^2)*(1 + x + x^2)).
(End)

A340445 Number of partitions of n into 3 parts that are not all the same.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 2, 4, 5, 6, 8, 10, 11, 14, 16, 18, 21, 24, 26, 30, 33, 36, 40, 44, 47, 52, 56, 60, 65, 70, 74, 80, 85, 90, 96, 102, 107, 114, 120, 126, 133, 140, 146, 154, 161, 168, 176, 184, 191, 200, 208, 216, 225, 234, 242, 252, 261, 270, 280, 290, 299, 310, 320, 330

Offset: 0

Wesley Ivan Hurt, Jan 07 2021

Conjecturally the same as A230059 (apart from the offset). - R. J. Mathar, Jan 14 2021

			a(6) = 2; [4,1,1], [3,2,1] ( [2,2,2] not counted ),
a(7) = 4; [5,1,1], [4,2,1], [3,3,1], [3,2,2],
a(8) = 5; [6,1,1], [5,2,1], [4,3,1], [4,2,2], [3,3,2],
a(9) = 6; [7,1,1], [6,2,1], [5,3,1], [4,4,1], [5,2,2], [4,3,2] ( [3,3,3] not counted ).

Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

Cf. A036410, A069905.

Table[Sum[Sum[(1 - KroneckerDelta[i, k, n - i - k]), {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 0, 80}]

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} (1 - [k = i = n-i-k]), where [ ] is the (generalized) Iverson bracket.
a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} (1 - [k = i] * [2*i = n-k] * [2*k = n-i]), where [ ] is the Iverson bracket.
From Alois P. Heinz, Jan 07 2021: (Start)
G.f.: x^4*(x^2-x-1)/((x+1)*(x^2+x+1)*(x-1)^3).
a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6), n>6. (End)
a(n) = A036410(n-1)-1. - Hugo Pfoertner, Jan 09 2021
a(n) + A079978(n) = A069905(n), n>0. - R. J. Mathar, Jan 18 2021
72*a(n) = -16*A099837(n+3) -9*(-1)^n +6*n^2 -31. - R. J. Mathar, Jun 09 2022

A348537 Number of partitions of n into 3 parts whose largest part divides n.

Original entry on oeis.org

0, 0, 1, 1, 0, 2, 0, 2, 1, 2, 0, 4, 0, 3, 1, 4, 0, 5, 0, 5, 1, 5, 0, 7, 0, 6, 1, 7, 0, 8, 0, 8, 1, 8, 0, 10, 0, 9, 1, 10, 0, 11, 0, 11, 1, 11, 0, 13, 0, 12, 1, 13, 0, 14, 0, 14, 1, 14, 0, 16, 0, 15, 1, 16, 0, 17, 0, 17, 1, 17, 0, 19, 0, 18, 1, 19, 0, 20, 0, 20, 1, 20, 0, 22, 0

Offset: 1

Wesley Ivan Hurt, Oct 21 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10480
Index entries for sequences related to partitions

Cf. A069905.

Array[Sum[Sum[(1 - Ceiling[#/(# - i - j)] + Floor[#/(# - i - j)]), {i, j, Floor[(# - j)/2]} ], {j, Floor[#/3]} ] &, 85] (* Michael De Vlieger, Oct 21 2021 *)

A348537(n) = sum(j=1,(n\3), sum(i=j,((n-j)\2), (1 - ceil(n/(n-i-j)) + floor(n/(n-i-j))))); \\ Antti Karttunen, Feb 18 2023

a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} (1 - ceiling(n/(n-i-j)) + floor(n/(n-i-j))).

A348538 Number of partitions of n into 3 parts whose smallest part divides n.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 3, 5, 5, 7, 5, 12, 6, 11, 12, 16, 8, 21, 9, 25, 18, 19, 11, 41, 18, 23, 24, 38, 14, 54, 15, 45, 30, 31, 36, 76, 18, 35, 36, 80, 20, 81, 21, 64, 68, 43, 23, 121, 39, 76, 48, 77, 26, 108, 60, 119, 54, 55, 29, 191, 30, 59, 101, 118, 72, 135, 33, 103, 66, 156, 35

Offset: 1

Wesley Ivan Hurt, Oct 21 2021

nonn

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for sequences related to partitions

Cf. A069905.

f:= proc(n) local i,j;
    add((floor((n-j)/2)-j+1), j = select(`<=`, numtheory:-divisors(n),n/3))
end proc:
map(f, [$1..100]); # Robert Israel, Sep 30 2024

Array[Sum[Sum[(1 - Ceiling[#/j] + Floor[#/j]), {i, j, Floor[(# - j)/2]}], {j, Floor[#/3]}] &, 71] (* Michael De Vlieger, Oct 21 2021 *)
Table[Count[IntegerPartitions[n,{3}],?(Divisible[n,#[[-1]]]&)],{n,80}] (* _Harvey P. Dale, Jul 10 2022 *)

a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} (1 - ceiling(n/j) + floor(n/j)).

A350043 Sum of all the parts > 1 in the partitions of n into 3 positive integer parts.

Original entry on oeis.org

0, 2, 7, 15, 24, 36, 58, 75, 104, 138, 175, 217, 277, 328, 399, 477, 560, 650, 766, 869, 1000, 1140, 1287, 1443, 1633, 1806, 2015, 2235, 2464, 2704, 2986, 3247, 3552, 3870, 4199, 4541, 4933, 5300, 5719, 6153, 6600, 7062, 7582, 8073, 8624, 9192, 9775, 10375, 11041, 11674

Offset: 3

Wesley Ivan Hurt, Dec 10 2021

			a(7) = 24; The partitions of 7 into 3 positive integer parts are (1,1,5), (1,2,4), (1,3,3) and (2,2,3). The sum of all the parts > 1 is then 5+2+4+3+3+2+2+3 = 24.

Winston de Greef, Table of n, a(n) for n = 3..10000
Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (0,2,2,-1,-4,-1,2,2,0,-1).

Cf. A069905, A350042.

CoefficientList[Series[-x*(x^9 - x^8 - 3*x^7 + 5*x^5 + 6*x^4 - 6*x^3 - 11*x^2 - 7*x - 2)/((x + 1)^2*(x^2 + x + 1)^2*(x - 1)^4), {x, 0, 50}], x] (* Wesley Ivan Hurt, Nov 12 2022 *)

a(n)=if(n==3, 0, -1 - floor((n-1)/2) + n * sum(k=1,floor(n/3), floor((n-3*k+2)/2))) \\ Winston de Greef, Jan 28 2024

For n >= 4, a(n) = -1 - floor((n-1)/2) + n * Sum_{k=1..floor(n/3)} floor((n-3*k+2)/2).
G.f.: -x^4 * (x^9-x^8-3*x^7+5*x^5+6*x^4-6*x^3-11*x^2-7*x-2) / ((x+1)^2 *(x^2+x+1)^2 *(x-1)^4). - Alois P. Heinz, Dec 13 2021
a(n) = 2*a(n-2)+2*a(n-3)-a(n-4)-4*a(n-5)-a(n-6)+2*a(n-7)+2*a(n-8)-a(n-10). - Wesley Ivan Hurt, Dec 17 2021

A382407 a(n) is the number of partitions n = x + y + z of positive integers such that xy + yz + x*z is a perfect square.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 3, 0, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 3, 0, 5, 1, 1, 2, 3, 3, 2, 1, 1, 3, 6, 1, 4, 2, 7, 4, 4, 0, 3, 5, 3, 4, 2, 1, 7, 2, 1, 5, 9, 3, 5, 3, 4, 1, 9, 2, 6, 3, 5, 6, 5, 4, 7, 5, 1, 5, 6, 3, 13, 7, 8, 4, 6, 0, 4, 4, 11, 5, 13, 2

Offset: 1

Felix Huber, Apr 04 2025

nonn

a(n) is the number of distinct cuboids with edge length 4*n whose surface area is half of a square.

Conjecture: a(k) = 0 iff k is an element of {2, 4, 8, 13} union A000244 union A005030.

			The a(14) = 3 partitions [x, y, z] are [1, 1, 12], [1, 4, 9] and [4, 4, 6] because 1*1 + 1*12 + 1*12 = 5^2, 1*4 + 4*9 + 1*9 = 7^2 and 4*4 + 4*6 + 4*6 = 8^2.

Felix Huber, Table of n, a(n) for n = 1..10000
Felix Huber, Maple program to calculate the partitions

Cf. A000244, A005030, A066955, A069905, A338939, A375512, A375576, A375580, A375731.

A382407:=proc(n)
    local a,x,y,z;
    a:=0;
    for x to n/3 do
        for y from x to (n-x)/2 do
            z:=n-x-y;
            if issqr(x*y+x*z+y*z) then
                a:=a+1
            fi
        od
    od;
    return a
end proc;
seq(A382407(n),n=1..87);

cp's OEIS Frontend

A350042 Sum of all the parts in the partitions of n into 3 positive integer parts.

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A360240 Weakly decreasing triples of positive integers sorted lexicographically and concatenated.

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A158138 Number of nondecreasing integer sequences of length 4 with sum zero and sum of absolute values 2n.

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AWK

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A328863 Number of partitions of 2n that describe the degree sequence of exactly one labeled multigraph with no loops.

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A338200 The number of similarity classes of pointed reflection spaces of residue two in an n-dimensional vector space over GF(2).

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Mathematica

PARI

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A340445 Number of partitions of n into 3 parts that are not all the same.

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Mathematica

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A348537 Number of partitions of n into 3 parts whose largest part divides n.

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Mathematica

PARI

Formula

A348538 Number of partitions of n into 3 parts whose smallest part divides n.

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Maple

A382407 a(n) is the number of partitions n = x + y + z of positive integers such that xy + yz + x*z is a perfect square.