A070770 -id:A070770 - cp's OEIS frontend

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

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

A070771 b+c+d+e where b>=c>=d>=e>=0 ordered by b then c then d then e.

Original entry on oeis.org

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

Offset: 0

Henry Bottomley, May 06 2002

nonn

Cf. A001477, A051162, A070770, A070772 for similar sequences with different numbers of terms summed.

A070772 b+c+d+e+f where b>=c>=d>=e>=f>=0 ordered by b then c then d then e then f.

Original entry on oeis.org

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

Offset: 0

Henry Bottomley, May 06 2002

nonn

Cf. A001477, A051162, A070770, A070771 for similar sequences with different numbers of terms summed.

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

A354388 Table read upward by antidiagonals: the n-th row gives the sums of each weakly decreasing nonnegative integer sequence of length n, listed in lexicographic order.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 2, 4, 0, 1, 2, 3, 3, 5, 0, 1, 2, 3, 2, 4, 6, 0, 1, 2, 3, 4, 3, 3, 7, 0, 1, 2, 3, 4, 2, 4, 4, 8, 0, 1, 2, 3, 4, 5, 3, 4, 5, 9, 0, 1, 2, 3, 4, 5, 2, 4, 5, 6, 10, 0, 1, 2, 3, 4, 5, 6, 3, 5, 6, 4, 11, 0, 1, 2, 3, 4, 5, 6, 2

Offset: 1

Peter Kagey, May 24 2022

			For n = 3, the weakly decreasing nonnegative integer sequences of length 3 listed in lexicographic order are (0,0,0), (1,0,0), (1,1,0), (1,1,1), (2,0,0), (2,1,0), (2,1,1), ....
The sums of these sequences correspond to row 3.
The table begins:
   1 | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, ...
   2 | 0, 1, 2, 2, 3, 4, 3, 4, 5, 6,  4,  5,  6,  7,  8, ...
   3 | 0, 1, 2, 3, 2, 3, 4, 4, 5, 6,  3,  4,  5,  5,  6, ...
   4 | 0, 1, 2, 3, 4, 2, 3, 4, 5, 4,  5,  6,  6,  7,  8, ...
   5 | 0, 1, 2, 3, 4, 5, 2, 3, 4, 5,  6,  4,  5,  6,  7, ...
   6 | 0, 1, 2, 3, 4, 5, 6, 2, 3, 4,  5,  6,  7,  4,  5, ...
   7 | 0, 1, 2, 3, 4, 5, 6, 7, 2, 3,  4,  5,  6,  7,  8, ...
   8 | 0, 1, 2, 3, 4, 5, 6, 7, 8, 2,  3,  4,  5,  6,  7, ...
   9 | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9,  2,  3,  4,  5,  6, ...
  10 | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,  2,  3,  4,  5, ...
  11 | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,  2,  3,  4, ...
  12 | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,  2,  3, ...
  ...| ...

Peter Kagey, Table of n, a(n) for n = 1..10011 (first 141 rows, flattened)

Cf. A001477 (row 1), A051162 (row 2, when read by rows), A070770 (row 3), A070771 (row 4), A070772 (row 5).

cp's OEIS Frontend

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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A070771 b+c+d+e where b>=c>=d>=e>=0 ordered by b then c then d then e.

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A070772 b+c+d+e+f where b>=c>=d>=e>=f>=0 ordered by b then c then d then e then f.

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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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A354388 Table read upward by antidiagonals: the n-th row gives the sums of each weakly decreasing nonnegative integer sequence of length n, listed in lexicographic order.

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