A056558 -id:A056558 - cp's OEIS frontend

A194849 Write n = C(i,3)+C(j,2)+C(k,1) with i>j>k>=0; let L[n] = [i,j,k]; sequence gives list of triples L[n], n >= 0.

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

Offset: 0

N. J. A. Sloane, Sep 03 2011

			List of triples begins:
[2, 1, 0]
[3, 1, 0]
[3, 2, 0]
[3, 2, 1]
[4, 1, 0]
[4, 2, 0]
[4, 2, 1]
[4, 3, 0]
[4, 3, 1]
[4, 3, 2]
...

D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.3, Eq. (20), p. 360.

The three columns are [A194847, A194848, A056558], or equivalently [A056556+2, A056557+1, A056558]. See A194847 for further information.

A127325 Hypertetrahedron with T(W,X,Y,Z) = Y - Z.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 1, 0, 0, 1, 0, 2, 1, 0, 3, 2, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 1, 0, 0, 1, 0, 2, 1, 0, 3, 2, 1, 0, 0, 1, 0, 2, 1, 0, 3, 2, 1, 0, 4, 3, 2, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 1, 0, 0, 1, 0, 2, 1, 0, 3, 2, 1, 0, 0, 1, 0, 2, 1, 0, 3, 2, 1, 0, 4, 3, 2, 1, 0

Offset: 0

Graeme McRae, Jan 10 2007

nonn

Together with A127324, A127326 and A127327 might enable reading "by antidiagonals" of hypercube arrays as 4-dimensional analog of A056558, A056560 and A056559 with cubical arrays.

			a(23)=1 because A127323(23) - A127324(23) = 1.
See A127327 for a table of A127324, A127325, A127326, A127327.

Cf. A127321, A127322, A127323, A127324, A127326, A127327.

Cf. A056558, A056560, A056559.

a(n) = A127323(n) - A127324(n).

A127326 Hypertetrahedron with T(W,X,Y,Z) = X - Y.

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 1, 1, 0, 0, 0, 0, 1, 0, 0, 2, 1, 1, 0, 0, 0, 3, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 1, 1, 0, 0, 0, 3, 2, 2, 1, 1, 1, 0, 0, 0, 0, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 1, 1, 0, 0, 0, 3, 2, 2, 1, 1, 1, 0, 0, 0, 0, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0

Offset: 0

Graeme McRae, Jan 10 2007

nonn

Together with A127324, A127325 and A127327 might enable reading "by antidiagonals" of hypercube arrays as 4-dimensional analog of A056558, A056560 and A056559 with cubical arrays.

			a(23)=0 because A127322(23) - A127323(23) = 0.
See A127327 for a table of A127324, A127325, A127326, A127327.

Cf. A127322, A127323, A127324, A127325, A127327.

Cf. A056558, A056560, A056559.

a(n) = A127322(n) - A127323(n).

A127327 Hypertetrahedron with T(W,X,Y,Z) = W - X.

Original entry on oeis.org

0, 1, 0, 0, 0, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 3, 3, 3, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 4, 4, 4, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Graeme McRae, Jan 10 2007

nonn

Together with A127324, A127325 and A127326 might enable reading "by antidiagonals" of hypercube arrays as 4-dimensional analog of A056558, A056560 and A056559 with cubical arrays.

			a(23)=1 because A127321(23) - A127322(23) = 1.
Table of A127324, A127325, A127326, A127327:
   n w,x,y,z
   0 0,0,0,0
   1 0,0,0,1
   2 0,0,1,0
   3 0,1,0,0
   4 1,0,0,0
   5 0,0,0,2
   6 0,0,1,1
   7 0,1,0,1
   8 1,0,0,1
   9 0,0,2,0
  10 0,1,1,0
  11 1,0,1,0
  12 0,2,0,0
  13 1,1,0,0
  14 2,0,0,0
  15 0,0,0,3
  16 0,0,1,2
  17 0,1,0,2
  18 1,0,0,2
  19 0,0,2,1
  20 0,1,1,1
  21 1,0,1,1
  22 0,2,0,1
  23 1,1,0,1

Cf. A127321, A127322, A127323, A127324, A127325, A127326.

Cf. A056558, A056560, A056559.

a(n) = A127321(n) - A127322(n).

A123575 The Kruskal-Macaulay function L_3(n).

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 2, 2, 3, 5, 5, 5, 6, 6, 7, 9, 9, 10, 12, 15, 15, 15, 16, 16, 17, 19, 19, 20, 22, 25, 25, 26, 28, 31, 35, 35, 35, 36, 36, 37, 39, 39, 40, 42, 45, 45, 46, 48, 51, 55, 55, 56, 58, 61, 65, 70, 70, 70, 71, 71, 72, 74, 74, 75, 77, 80, 80, 81, 83, 86, 90, 90, 91, 93

Offset: 0

N. J. A. Sloane, Nov 12 2006

Write n (uniquely) as n = C(n_t,t) + C(n_{t-1},t-1) + ... + C(n_v,v) where n_t > n_{t-1} > ... > n_v >= v >= 1. Then L_t(n) = C(n_t,t+1) + C(n_{t-1},t) + ... + C(n_v,v+1).

D. E. Knuth, The Art of Computer Programming, Vol. 4, Fascicle 3, Section 7.2.1.3, Table 3.

For L_i(n), i=1, 2, 3, 4, 5 see A000217, A111138, A123575, A123576, A123577.

Essentially partial sums of A056558.

lowpol := proc(n,t) local x::integer ; x := floor( (n*factorial(t))^(1/t)) ; while binomial(x,t) <= n do x := x+1 ; od ; RETURN(x-1) ; end: C := proc(n,t) local nresid,tresid,m,a ; nresid := n ; tresid := t ; a := [] ; while nresid > 0 do m := lowpol(nresid,tresid) ; a := [op(a),m] ; nresid := nresid - binomial(m,tresid) ; tresid := tresid-1 ; od ; RETURN(a) ; end: L := proc(n,t) local a ; a := C(n,t) ; #add( binomial(op(i,a),t+i),i=1..nops(a)) ; add( binomial(op(i,a),t+2-i),i=1..nops(a)) ; end: A123575 := proc(n) L(n,3) ; end: for n from 0 to 80 do printf("%d, ",A123575(n)) ; od ; # R. J. Mathar, May 18 2007

lowpol[n_, t_] := Module[{x = Floor[(n t!)^(1/t)]}, While[Binomial[x, t] <= n, x++] ; x - 1];
c[n_, t_] := Module[{n0 = n, t0 = t, m, a = {}}, While[n0 > 0, m = lowpol[n0, t0]; AppendTo[a, m]; n0 -= Binomial[m, t0]; t0--]; a];
L[n_, t_] := Module[{a = c[n, t]}, Sum[Binomial[a[[i]], t + 2 - i], {i, 1, Length[a]}]];
a[n_] := L[n, 3];
a /@ Range[0, 80] (* Jean-François Alcover, Mar 29 2020, after R. J. Mathar *)

More terms from R. J. Mathar, May 18 2007

A309362 Positions of 0's in A326764 (interpreted as a flat sequence).

Original entry on oeis.org

0, 11, 15, 17, 37, 45, 54, 59, 70, 79, 124, 129, 135, 161, 171, 192, 195, 202, 213, 252, 272, 299, 306, 307, 358, 372, 410, 422, 477, 486, 498, 506, 530, 571, 586, 644, 655, 736, 749, 760, 794, 828, 845, 890, 905, 939, 985, 994, 1087, 1101, 1113, 1168, 1189

Offset: 0

Rémy Sigrist, Jul 25 2019

nonn

Rémy Sigrist, Table of n, a(n) for n = 0..9964
Rémy Sigrist, PARI program for A309362

Cf. A056558, A056559, A056560, A326757, A326758, A326759, A326764.

PARI
```
See Links section.
```

A326757(n) = A056558(a(n)).
A326758(n) = A056560(a(n)).
A326759(n) = A056559(a(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.

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

Original entry on oeis.org

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

cp's OEIS Frontend

A194849 Write n = C(i,3)+C(j,2)+C(k,1) with i>j>k>=0; let L[n] = [i,j,k]; sequence gives list of triples L[n], n >= 0.

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A127325 Hypertetrahedron with T(W,X,Y,Z) = Y - Z.

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A127326 Hypertetrahedron with T(W,X,Y,Z) = X - Y.

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A127327 Hypertetrahedron with T(W,X,Y,Z) = W - X.

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A123575 The Kruskal-Macaulay function L_3(n).

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A309362 Positions of 0's in A326764 (interpreted as a flat sequence).

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

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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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