A331195 -id:A331195 - cp's OEIS frontend

A332616 a(n) = value of the cubic form A^3 + B^3 + C^3 - 3ABC evaluated at row n of the table in A331195.

0, 1, 2, 0, 8, 9, 4, 16, 5, 0, 27, 28, 20, 35, 18, 7, 54, 28, 8, 0, 64, 65, 54, 72, 49, 32, 91, 56, 27, 10, 128, 81, 40, 11, 0, 125, 126, 112, 133, 104, 81, 152, 108, 70, 44, 189, 130, 77, 36, 13, 250, 176, 108, 52, 14, 0, 216, 217, 200, 224, 189, 160, 243

Offset: 0

Mehmet A. Ates, Jun 08 2020

No term in the sequence is congruent to 3 or 6 (mod 9).

			For n=3, a(n) = f[1,1,0] = 1^3 + 1^3 + 0^3 - 3*1*1*0 = 2.

Mehmet A. Ates, Table of n, a(n) for n = 0..19599
Mathematical Association of America, 2019 William Lowell Putnam Mathematical Competition Problems

Cf. A056556, A056557, A056558.
Cf. A330013, A331195.
Cf. A074232 (in ascending order, strictly positive & without duplicates).

SeqSize = 30;
ListSize = 120;
F3List = List[];
f3[a_, b_, c_] := a^3 + b^3 + c^3 - 3*a*b*c
For[i = 0, i <= SeqSize, i++, For[j = 0, j <= i, j++, For[k = 0, k <= j, k++, AppendTo[F3List, f3[i, j, k]]]]]
ListPlot[F3List, PlotLabel -> "a(n)"]
Print["First ", ListSize, " elements of a(n): ", Take[F3List, ListSize]]

a(n) = A056556(n)^3 + A056557(n)^3 + A056558(n)^3 - 3*A056556(n)*A056557(n)*A056558(n).

Edited by N. J. A. Sloane, Aug 06 2020

A360071 Regular tetrangle where T(n,k,i) = number of integer partitions of n of length k with i distinct parts.

Original entry on oeis.org

1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 2, 1, 1, 1, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 3, 0, 3, 1, 0, 2, 1, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Jan 28 2023

I call this a tetrangle because it is a sequence of finite triangles. - Gus Wiseman, Jan 30 2023

			Tetrangle begins:
  1   1     1       1         1           1             1
      1 0   0 1     1 1       0 2         1 2           0 3
            1 0 0   0 1 0     0 2 0       1 1 1         0 3 1
                    1 0 0 0   0 1 0 0     0 2 0 0       0 2 1 0
                              1 0 0 0 0   0 1 0 0 0     0 2 0 0 0
                                          1 0 0 0 0 0   0 1 0 0 0 0
                                                        1 0 0 0 0 0 0
For example, finite triangle n = 5 counts the following partitions:
    (5)
     .    (41)(32)
     .   (311)(221)  .
     .     (2111)    .   .
  (11111)     .      .   .   .

Row sums are A008284 (partitions by number of parts), reverse A058398.
First columns i = 1 are A051731.
Last columns i = k are A060016.
Column sums are A116608 (partitions by number of distinct parts).
Positive terms are counted by A360072.
A000041 counts partitions, strict A000009.
Other tetrangles: A318393, A318816, A320808, A334433, A345197.
Cf. A055884, A327482, A331195, A360010, A360069.

Table[Length[Select[IntegerPartitions[n], Length[#]==k&&Length[Union[#]]==i&]],{n,1,9},{k,1,n},{i,1,k}]

A360010 First part of the n-th weakly decreasing triple of positive integers sorted lexicographically. Each n > 0 is repeated A000217(n) times.

Original entry on oeis.org

1, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8

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), ...

For pairs instead of triples we have A002024.
Positions of first appearances are A050407(n+2) = A000292(n)+1.
The zero-based version is A056556.
The triples have sums A070770.
The second instead of first part is A194848.
The third instead of first part is A333516.
Concatenating all the triples gives A360240.
Cf. A000217, A003056, A056557, A056558, A069905, A158842, A331195.

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

from math import comb
from sympy import integer_nthroot
def A360010(n): return (m:=integer_nthroot(6*n,3)[0])+(n>comb(m+2,3)) # Chai Wah Wu, Nov 04 2024

a(n) = A056556(n) + 1 = A331195(3n) + 1.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/8 + log(2)/4. - Amiram Eldar, Feb 18 2024
a(n) = m+1 if n>binomial(m+2,3) and a(n) = m otherwise where m = floor((6n)^(1/3)). - Chai Wah Wu, Nov 04 2024

A330709 Two-column table read by rows: pairs (i,j) in order sorted from the left.

Original entry on oeis.org

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

Offset: 0

Mehmet A. Ates, Jun 08 2020

Mehmet A. Ates, Table of n, a(n) for n = 0..1500

a(2n)= A003056(n),
a(2n+1)= A002262(n).
Cf. A331195.

TwoDVectors = List[];
SeqSize = 20;
For[i = 0, i <= SeqSize, i++,
  For[j = 0, j <= i, j++,
    AppendTo[TwoDVectors, {i, j}]
  ]
];
Flatten[TwoDVectors]

A360072 Number of pairs of positive integers (k,i) such that k >= i and there exists an integer partition of n of length k with i distinct parts.

Original entry on oeis.org

0, 1, 2, 3, 5, 5, 9, 9, 13, 14, 18, 19, 26, 25, 30, 34, 39, 40, 48, 48, 56, 59, 64, 67, 78, 78, 84, 89, 97, 99, 111, 111, 121, 125, 131, 137, 149, 149, 158, 165, 176, 177, 190, 191, 202, 210, 216, 222, 238, 239, 250, 256, 266, 270, 284, 289, 302, 307, 316, 323

Offset: 0

Gus Wiseman, Jan 28 2023

nonn

This is the number of nonzero terms in the n-th triangle of A360071.

			The a(5) = 5 pairs are: (1,1), (2,2), (3,2), (4,2), (5,1). The pair (3,3) is absent because it is not possible to partition 5 into 3 parts, all 3 of which are distinct.
The a(6) = 9 pairs are: (1,1), (2,1), (2,2), (3,1), (3,2), (3,3), (4,2), (5,2), (6,1). The pair (3,3) is present because (3,2,1) is a partition of 6 into 3 parts, all 3 of which are distinct.

Andrew Howroyd, Table of n, a(n) for n = 0..1000

A000041 counts integer partitions, strict A000009.
A008284 counts partitions by number of parts, reverse A058398.
A116608 counts partitions by number of distinct parts.
Cf. A000005, A051731, A055884, A060016, A331195, A360010, A360071.

Table[Count[Flatten[Sign[Table[Length[Select[IntegerPartitions[n], Length[#]==k&&Length[Union[#]]==i&]],{k,1,n},{i,1,k}]]],1],{n,0,30}]

a(n) = if(n < 1, 0, numdiv(n) + sum(k=2, (sqrtint(8*n+1)-1)\2, n-binomial(k+1,2)+1)) \\ Andrew Howroyd, Jan 30 2023

a(n) = A000005(n) + Sum_{k=2..floor((sqrt(8*n+1)-1)/2)} (1 + n - binomial(k+1,2)) for n > 0. - Andrew Howroyd, Jan 30 2023

Terms a(31) and beyond from Andrew Howroyd, Jan 30 2023

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.

A372667 Norm i^2+j^2+k^2 of (i,j,k) for 0 <= k <= j <= i.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 9, 12, 9, 10, 11, 13, 14, 17, 18, 19, 22, 27, 16, 17, 18, 20, 21, 24, 25, 26, 29, 34, 32, 33, 36, 41, 48, 25, 26, 27, 29, 30, 33, 34, 35, 38, 43, 41, 42, 45, 50, 57, 50, 51, 54, 59, 66, 75, 36, 37, 38, 40, 41, 44, 45, 46, 49, 54, 52, 53

Offset: 0

A. Timothy Royappa, May 09 2024

In crystallography, these triples (i,j,k) can be interpreted as Miller indices, which can be sorted into a list: (0 0 0), (1 0 0), (1 1 0), (1 1 1), (2 0 0), (2 1 0), (2 1 1), (2 2 0), (2 2 1), (2 2 2), (3 0 0), (3 1 0), (3 1 1), (3 2 0), etc.

			The first few triples are:
   0, 0, 0
   1, 0, 0
   1, 1, 0
   1, 1, 1
   2, 0, 0
   2, 1, 0
   2, 1, 1
   2, 2, 0
   2, 2, 1
   2, 2, 2
   3, 0, 0
   ...

C. Suryanarayana and M. Grant Norton, X-Ray Diffraction - A Practical Approach, Springer Science + Business Media, 1998, p. 83.

Robert Israel, Table of n, a(n) for n = 0..10000
S. Phil Ahrenkiel, List of the first few Miller indices, in order, excluding (0 0 0)

The table of triples forms A331195.

Cf. A070770, A069011 (2-dimensional analog), A004215 (complement to this sequence)

a:=[];
for i from 0 to 10 do for j from 0 to i do for k from 0 to j do
a:=[op(a),i^2+j^2+k^2]; od: od: od: a; # N. J. A. Sloane, Jun 03 2024

print([i**2 + j**2 + k**2 for i in range(7) for j in range(i+1) for k in range(j+1)]) # Andrey Zabolotskiy, May 09 2024

More terms from Andrey Zabolotskiy, May 09 2024

cp's OEIS Frontend

A332616 a(n) = value of the cubic form A^3 + B^3 + C^3 - 3ABC evaluated at row n of the table in A331195.

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A360071 Regular tetrangle where T(n,k,i) = number of integer partitions of n of length k with i distinct parts.

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A360010 First part of the n-th weakly decreasing triple of positive integers sorted lexicographically. Each n > 0 is repeated A000217(n) times.

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A330709 Two-column table read by rows: pairs (i,j) in order sorted from the left.

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A360072 Number of pairs of positive integers (k,i) such that k >= i and there exists an integer partition of n of length k with i distinct parts.

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

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A372667 Norm i^2+j^2+k^2 of (i,j,k) for 0 <= k <= j <= i.

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