A056558 -id:A056558 - cp's OEIS frontend

A127324 Fourth 4-dimensional hyper-tetrahedral coordinate; 4-D analog of A056558.

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

Offset: 0

Graeme McRae, Jan 10 2007

nonn

Alternatively, write n = C(i,4)+C(j,3)+C(k,2)+C(l,1) with i>j>k>l>=0; sequence gives k values. Each n >= 0 has a unique representation as n = C(i,4)+C(j,3)+C(k,2)+C(l.1) with i>j>k>l>=0. This is the combinatorial number system of degree t = 4, where we get [A194882, A194883, A194884, A127324].
If {(W,X,Y,Z)} are 4-tuples of nonnegative integers with W>=X>=Y>=Z ordered by W, X, Y and Z, then W=A127321(n), X=A127322(n), Y=A127323(n) and Z=A127324(n). These sequences are the four-dimensional analogs of the three-dimensional A056556, A056557 and A056558.
This is a 'Matryoshka doll' sequence with alpha=0 (cf. A055462 and A000332), seq(seq(seq(seq(i,i=alpha..k),k=alpha..n),n=alpha..m),m=alpha..4). - Peter Luschny, Jul 14 2009

			See A127321 for a table of A127321, A127322, A127323, A127324
See A127327 for a table of A127324, A127325, A127326, A127327

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

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A127321, A127322, A127323, A056556, A056557, A056558, A000332, A000292, A000217.

Cf. A002262, A056558.

import Data.List (inits)
a127324 n = a127324_list !! n
a127324_list = concatMap (concatMap concat .
               inits . inits . enumFromTo 0) $ enumFrom 0
-- Reinhard Zumkeller, Jun 01 2015

seq(seq(seq(seq(i,i=0..k),k=0..n),n=0..m),m=0..5); # Peter Luschny, Sep 22 2011

Table[i, {m, 0, 5}, {k, 0, m}, {j, 0, k}, {i, 0, j}] // Flatten  (* Robert G. Wilson v, Sep 27 2011 *)

For W>=X>=Y>=Z>=0, a(A000332(W+3)+A000292(X)+A000217(Y)+Z) = Z A127322(n+1) = A127321(n)==A127324(n) ? 0 : A127322(n)==A127324(n) ? 0 : A127323(n)==A127324(n) ? 0 : A127324(n)+1

A127560 Number of fixed r-celled polyominoes with smallest containing rectangle measuring k by m, read in order r=A056556(n)+1, k=A056560(n)+1, m=A056558(n)+1.

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 0, 1, 4, 1, 0, 0, 0, 0, 1, 0, 1, 8, 8, 1, 0, 0, 0, 0, 0, 0, 0, 6, 6, 0, 1, 12, 25, 12, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 18, 44, 18, 0, 1, 16, 50, 50, 16, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 32, 8, 0, 0, 38, 155, 155, 38, 0, 1, 20, 82, 120, 82, 20, 1

Offset: 0

Graeme McRae, Jan 18 2007

nonn

The sum of each triangle, i.e. for a given r the sum of a(n) for all n such that r=A056556(n)+1 is the number of r-celled fixed polyominoes, A001168(r).

			The 5th triangle of the sequence, the number of fixed pentominoes by dimension, is
0,0,0,0,1
0,0,6,12
0,6,25
0,12
1
This indicates, for example, there are 25 fixed pentominos that fit in a 3 X 3 rectangle and 12 fixed pentominos that fit in a 4 X 2 rectangle.

Cf. A000105, A000988, A001168 Indices for reading by triangles given by A056556, A056560, A056558.

A002262 Triangle read by rows: T(n,k) = k, 0 <= k <= n, in which row n lists the first n+1 nonnegative integers.

Original entry on oeis.org

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

Offset: 0

Angele Hamel (amh(AT)maths.soton.ac.uk)

The point with coordinates (x = A025581(n), y = A002262(n)) sweeps out the first quadrant by upwards antidiagonals. N. J. A. Sloane, Jul 17 2018
Old name: Integers 0 to n followed by integers 0 to n+1 etc.
a(n) = n - the largest triangular number <= n. - Amarnath Murthy, Dec 25 2001
The PARI functions t1, t2 can be used to read a square array T(n,k) (n >= 0, k >= 0) by antidiagonals downwards: n -> T(t1(n), t2(n)). - Michael Somos, Aug 23 2002
Values x of unique solution pair (x,y) to equation T(x+y) + x = n, where T(k)=A000217(k). - Lekraj Beedassy, Aug 21 2004
a(A000217(n)) = 0; a(A000096(n)) = n. - Reinhard Zumkeller, May 20 2009
Concatenation of the set representation of ordinal numbers, where the n-th ordinal number is represented by the set of all ordinals preceding n, 0 being represented by the empty set. - Daniel Forgues, Apr 27 2011
An integer sequence is nonnegative if and only if it is a subsequence of this sequence. - Charles R Greathouse IV, Sep 21 2011
a(A195678(n)) = A000040(n) and a(m) <> A000040(n) for m < A195678(n), an example of the preceding comment. - Reinhard Zumkeller, Sep 23 2011
A sequence B is called a reluctant sequence of sequence A, if B is triangle array read by rows: row number k coincides with first k elements of the sequence A. A002262 is reluctant sequence of 0,1,2,3,... The nonnegative integers, A001477. - Boris Putievskiy, Dec 12 2012

			From _Daniel Forgues_, Apr 27 2011: (Start)
Examples of set-theoretic representation of ordinal numbers:
  0: {}
  1: {0} = {{}}
  2: {0, 1} = {0, {0}} = {{}, {{}}}
  3: {0, 1, 2} = {{}, {0}, {0, 1}} = ... = {{}, {{}}, {{}, {{}}}} (End)
From _Omar E. Pol_, Jul 15 2012: (Start)
  0;
  0, 1;
  0, 1, 2;
  0, 1, 2, 3;
  0, 1, 2, 3, 4;
  0, 1, 2, 3, 4, 5;
  0, 1, 2, 3, 4, 5, 6;
  0, 1, 2, 3, 4, 5, 6, 7;
  0, 1, 2, 3, 4, 5, 6, 7, 8;
  0, 1, 2, 3, 4, 5, 6, 7, 8, 9;
  0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10;
(End)

Charles R Greathouse IV, Rows n = 0..100, flattened
Boris Putievskiy, Transformations [Of] Integer Sequences And Pairing Functions, arXiv preprint arXiv:1212.2732 [math.CO], 2012.
Michael Somos, Sequences used for indexing triangular or square arrays

Cf. A002024, A002260, A004736, A025581, A025675, A025682.
Cf. A025691, A048645, A053186, A053645, A056558, A127324.
As a sequence, essentially same as A048151.
Cf. A060510 (parity).

a002262 n k = a002262_tabl !! n !! k
a002262_row n = a002262_tabl !! n
a002262_tabl = map (enumFromTo 0) [0..]
a002262_list = concat a002262_tabl
-- Reinhard Zumkeller, Aug 05 2015, Jul 13 2012, Mar 07 2011

seq(seq(i,i=0..n),n=0..14); # Peter Luschny, Sep 22 2011
A002262 := n -> n - binomial(floor((1/2)+sqrt(2*(1+n))),2);

m[n_]:= Floor[(-1 + Sqrt[8n - 7])/2]
b[n_]:= n - m[n] (m[n] + 1)/2
Table[m[n], {n, 1, 105}]     (* A003056 *)
Table[b[n], {n, 1, 105}]     (* A002260 *)
Table[b[n] - 1, {n, 1, 120}] (* A002262 *)
(* Clark Kimberling, Jun 14 2011 *)
Flatten[Table[k, {n, 0, 14}, {k, 0, n}]] (* Alonso del Arte, Sep 21 2011 *)
Flatten[Table[Range[0,n], {n,0,15}]] (* Harvey P. Dale, Jan 31 2015 *)

PARI
```
a(n)=n-binomial(round(sqrt(2+2*n)),2)
```

t1(n)=n-binomial(floor(1/2+sqrt(2+2*n)),2) /* A002262, this sequence */

t2(n)=binomial(floor(3/2+sqrt(2+2*n)),2)-(n+1) /* A025581, cf. comment by Somos for reading arrays by antidiagonals */

concat(vector(15,n,vector(n,i,i-1)))  \\ M. F. Hasler, Sep 21 2011

apply( {A002262(n)=n-binomial(sqrtint(8*n+8)\/2,2)}, [0..99]) \\ M. F. Hasler, Oct 20 2022

for i in range(16):
    for j in range(i):
        print(j, end=", ") # Mohammad Saleh Dinparvar, May 13 2020

from math import comb, isqrt
def a(n): return n - comb((1+isqrt(8+8*n))//2, 2)
print([a(n) for n in range(105)]) # Michael S. Branicky, May 07 2023

a(n) = A002260(n) - 1.
a(n) = n - (trinv(n)*(trinv(n)-1))/2; trinv := n -> floor((1+sqrt(1+8*n))/2) (cf. A002024); # gives integral inverses of triangular numbers
a(n) = n - A000217(A003056(n)) = n - A057944(n). - Lekraj Beedassy, Aug 21 2004
a(n) = A140129(A023758(n+2)). - Reinhard Zumkeller, May 14 2008
a(n) = f(n,1) with f(n,m) = if nReinhard Zumkeller, May 20 2009
a(n) = (1/2)*(t - t^2 + 2*n), where t = floor(sqrt(2*n+1) + 1/2) = round(sqrt(2*n+1)). - Ridouane Oudra, Dec 01 2019
a(n) = ceiling((-1 + sqrt(9 + 8*n))/2) * (1 - ((1/2)*ceiling((1 + sqrt(9 + 8*n))/2))) + n. - Ryan Jean, Sep 03 2022
G.f.: x*y/((1 - x)*(1 - x*y)^2). - Stefano Spezia, Feb 21 2024

New name from Omar E. Pol, Jul 15 2012

Typo in definition fixed by Reinhard Zumkeller, Aug 05 2015

A014311 Numbers with exactly 3 ones in binary expansion.

Original entry on oeis.org

7, 11, 13, 14, 19, 21, 22, 25, 26, 28, 35, 37, 38, 41, 42, 44, 49, 50, 52, 56, 67, 69, 70, 73, 74, 76, 81, 82, 84, 88, 97, 98, 100, 104, 112, 131, 133, 134, 137, 138, 140, 145, 146, 148, 152, 161, 162, 164, 168, 176, 193, 194, 196, 200, 208, 224, 259, 261, 262, 265, 266, 268, 273, 274, 276, 280, 289, 290, 292, 296, 304

Offset: 1

Al Black (gblack(AT)nol.net)

Equivalently, sums of three distinct powers of 2.
Appears to give all n such that 64 is the highest power of 2 dividing A005148(n). - Benoit Cloitre, Jun 22 2002
From Gus Wiseman, Oct 05 2020: (Start)
These are numbers k such that the k-th composition in standard order has length 3. The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions. The sequence together with the corresponding standard compositions begins:
(1,1,1)     44: (2,1,3)     97: (1,5,1)
(2,1,1)     49: (1,4,1)     98: (1,4,2)
(1,2,1)     50: (1,3,2)    100: (1,3,3)
(1,1,2)     52: (1,2,3)    104: (1,2,4)
(3,1,1)     56: (1,1,4)    112: (1,1,5)
(2,2,1)     67: (5,1,1)    131: (6,1,1)
(2,1,2)     69: (4,2,1)    133: (5,2,1)
(1,3,1)     70: (4,1,2)    134: (5,1,2)
(1,2,2)     73: (3,3,1)    137: (4,3,1)
(1,1,3)     74: (3,2,2)    138: (4,2,2)
(4,1,1)     76: (3,1,3)    140: (4,1,3)
(3,2,1)     81: (2,4,1)    145: (3,4,1)
(3,1,2)     82: (2,3,2)    146: (3,3,2)
(2,3,1)     84: (2,2,3)    148: (3,2,3)
(2,2,2)     88: (2,1,4)    152: (3,1,4)
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Robert Baillie, Summing the curious series of Kempner and Irwin, arXiv:0806.4410 [math.CA], 2008-2015. See p. 18 for Mathematica code irwinSums.m.
Stephen Morley, HAKMEM Item 175 (Gosper).
Tilman Piesk, First 56 elements in a tetrahedral array.

Cf. A038465 (base 3), A038471 (base 4), A038475 (base 5).
Cf. A081091 (primes), A212190 (squares), A212192 (triangular numbers), A173589 (Fibbinary).
Cf. A057168.
Cf. A000079, A018900, A014311, A014312, A014313, A023688, A023689, A023690, A023691 (Hammingweight = 1, 2, ..., 9).
Cf. A056558, A194848, A194847.
A000217(n-2) counts compositions into three parts.
A001399(n-3) = A069905(n) = A211540(n+2) counts the unordered case.
A001399(n-6) = A069905(n-3) = A211540(n-1) counts the unordered strict case.
A001399(n-6)*6 = A069905(n-3)*6 = A211540(n-1)*6 counts the strict case.
A014612 is an unordered version, with strict case A007304.
A337453 is the strict case.
A337461 counts the coprime case.
A033992 lists numbers divisible by exactly three different primes.
A323024 lists numbers with exactly three different prime multiplicities.
Cf. A220377, A307534, A337459, A337460, A337561, A337603, A337604.

unsigned hakmem175(unsigned x) {
    unsigned s, o, r;
    s = x & -x;  r = x + s;
    o = r ^ x;  o = (o >> 2) / s;
    return r | o;
}
unsigned A014311(int n) {
    if (n == 1) return 7;
    return hakmem175(A014311(n - 1));
}  // Peter Luschny, Jan 01 2014

a014311 n = a014311_list !! (n-1)
a014311_list = [2^x + 2^y + 2^z |
                x <- [2..], y <- [1..x-1], z <- [0..y-1]]
-- Reinhard Zumkeller, May 03 2012

Select[Range[200], (Count[IntegerDigits[#, 2], 1] == 3)&]
nn = 8; Flatten[Table[2^i + 2^j + 2^k, {i, 2, nn}, {j, 1, i - 1}, {k, 0, j - 1}]] (* T. D. Noe, Nov 05 2013 *)

for(n=0,10^3,if(hammingweight(n)==3,print1(n,", "))); \\ Joerg Arndt, Mar 04 2014

print1(t=7);for(i=2,50,print1(","t=A057168(t))) \\ M. F. Hasler, Aug 27 2014

A014311_list = [2**a+2**b+2**c for a in range(2,6) for b in range(1,a) for c in range(b)] # Chai Wah Wu, Jan 24 2021

from itertools import islice
def A014311_gen(): # generator of terms
    yield (n:=7)
    while True: yield (n:=n^((a:=-n&n+1)|(a>>1)) if n&1 else ((n&~(b:=n+(a:=n&-n)))>>a.bit_length())^b)
A014311_list = list(islice(A014311_gen(),20)) # Chai Wah Wu, Mar 10 2025

from math import isqrt, comb
from sympy import integer_nthroot
def A014311(n): return (1<<(r:=n-1-comb((m:=integer_nthroot(6*n,3)[0])+(t:=(n>comb(m+2,3)))+1,3))-comb((k:=isqrt(b:=r+1<<1))+(b>k*(k+1)),2))+(1<<(a:=isqrt(s:=n-comb(m-(t^1)+2,3)<<1))+((s<<2)>(a<<2)*(a+1)+1))+(1<Chai Wah Wu, Mar 10 2025

A000120(a(n)) = 3. - Reinhard Zumkeller, May 03 2012
Start with A084468. If n is in sequence, then 2n is too. - Ralf Stephan, Aug 16 2013
a(n+1) = A057168(a(n)). - M. F. Hasler, Aug 27 2014
a(n) = 2^A056558(n-1) + 2^A194848(n-1) + 2^A194847(n-1). - Ridouane Oudra, Sep 06 2020
Sum_{n>=1} 1/a(n) = A367110 = 1.428591545852638123996854844400537952781688750906133068397189529775365950039... (calculated using Baillie's irwinSums.m, see Links). - Amiram Eldar, Feb 14 2022

Extension and program by Olivier Gérard

A052217 Numbers whose sum of digits is 3.

Original entry on oeis.org

3, 12, 21, 30, 102, 111, 120, 201, 210, 300, 1002, 1011, 1020, 1101, 1110, 1200, 2001, 2010, 2100, 3000, 10002, 10011, 10020, 10101, 10110, 10200, 11001, 11010, 11100, 12000, 20001, 20010, 20100, 21000, 30000, 100002, 100011, 100020, 100101

Offset: 1

Henry Bottomley, Feb 01 2000

From Joshua S.M. Weiner, Oct 19 2012: (Start)
Sequence is a representation of the "energy states" of "multiplex" notation of 3 quantum of objects in a juggling pattern.
0 = an empty site, or empty hand. 1 = one object resides in the site. 2 = two objects reside in the site. 3 = three objects reside in the site. (See A038447.) (End)
A007953(a(n)) = 3; number of repdigits = #{3,111} = A242627(3) = 2. - Reinhard Zumkeller, Jul 17 2014
Can be seen as a table whose n-th row holds the n-digit terms {10^(n-1) + 10^m + 10^k, 0 <= k <= m < n}, n >= 1. Row lengths are then (1, 3, 6, 10, ...) = n*(n+1)/2 = A000217(n). The first and the n last terms of row n are 10^(n-1) + 2 resp. 2*10^(n-1) + 10^k, 0 <= k < n. - M. F. Hasler, Feb 19 2020

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (terms 1..84 from Vincenzo Librandi, terms 85..1140 from T. D. Noe)

Cf. A069521 to A069530, A069532 to A069537.
Cf. A007953, A218043 (subsequence).
Row n=3 of A245062.
Other digit sums: A011557 (1), A052216 (2), A052218 (4), A052219 (5), A052220 (6), A052221 (7), A052222 (8), A052223 (9), A052224 (10), A166311 (11), A235151 (12), A143164 (13), A235225(14), A235226 (15), A235227 (16), A166370 (17), A235228 (18), A166459 (19), A235229 (20).
Other bases: A014311 (binary), A226636 (ternary), A179243 (Zeckendorf).
Cf. A242614, A242627.
Cf. A003056, A002262 (triangular coordinates), A056556, A056557, A056558 (tetrahedral coordinates).

a052217 n = a052217_list !! (n-1)
a052217_list = filter ((== 3) . a007953) [0..]
-- Reinhard Zumkeller, Jul 17 2014

[n: n in [1..100101] | &+Intseq(n) eq 3 ]; // Vincenzo Librandi, Mar 07 2013

Union[FromDigits/@Select[Flatten[Table[Tuples[Range[0,3],n],{n,6}],1],Total[#]==3&]] (* Harvey P. Dale, Oct 20 2012 *)
Select[Range[10^6], Total[IntegerDigits[#]] == 3 &] (* Vincenzo Librandi, Mar 07 2013 *)
Union[Flatten[Table[FromDigits /@ Permutations[PadRight[s, 18]], {s, IntegerPartitions[3]}]]] (* T. D. Noe, Mar 08 2013 *)

isok(n) = sumdigits(n) == 3; \\ Michel Marcus, Dec 28 2015

apply( {A052217_row(n,s,t=-1)=vector(n*(n+1)\2,k,t++>s&&t=!s++;10^(n-1)+10^s+10^t)}, [1..5]) \\ M. F. Hasler, Feb 19 2020

from itertools import count, islice
def agen(): yield from (10**i + 10**j + 10**k for i in count(0) for j in range(i+1) for k in range(j+1))
print(list(islice(agen(), 40))) # Michael S. Branicky, May 14 2022

from math import comb, isqrt
from sympy import integer_nthroot
def A052217(n): return 10**((m:=integer_nthroot(6*n,3)[0])-(a:=n<=comb(m+2,3)))+10**((k:=isqrt(b:=(c:=n-comb(m-a+2,3))<<1))-((b<<2)<=(k<<2)*(k+1)+1))+10**(c-1-comb(k+(b>k*(k+1)),2)) # Chai Wah Wu, Dec 11 2024

T(n,k) = 10^(n-1) + 10^A003056(k) + 10^A002262(k) when read as a table with row lengths n*(n+1)/2, n >= 1, 0 <= k < n*(n+1)/2. - M. F. Hasler, Feb 19 2020

a(n) = 10^A056556(n-1) + 10^A056557(n-1) + 10^A056558(n-1). - Kevin Ryde, Apr 17 2021

Offset changed from 0 to 1 by Vincenzo Librandi, Mar 07 2013

A056556 First tetrahedral coordinate; repeat m (m+1)*(m+2)/2 times.

Original entry on oeis.org

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

Offset: 0

Henry Bottomley, Jun 26 2000

If {(X,Y,Z)} are triples of nonnegative integers with X >= Y >= Z ordered by X, Y and Z, then X=A056556(n), Y=A056557(n) and Z=A056558(n).
From Gus Wiseman, Jul 03 2019: (Start)
Also the maximum number of distinct multiplicities among integer partitions of n. For example, random partitions of 56 realizing each number of distinct multiplicities are:
  1: (24,17,6,5,3,1)
  2: (10,9,9,5,5,4,4,3,3,2,1,1)
  3: (6,5,5,5,4,4,4,3,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1)
  4: (28,5,5,3,3,3,2,2,1,1,1,1,1)
  5: (13,4,4,4,4,4,3,3,3,2,2,2,2,2,2,1,1)
  6: (6,5,5,4,4,4,3,3,3,3,2,2,2,2,2,1,1,1,1,1,1)
The maximum number of distinct multiplicities is 6, so a(56) = 6.
(End)

			3 is (3+1) * (3+2)/2 = 10 times in the sequence all these occurrences are in consecutive places. The first 3 is at position binomial(3 + 2, 3) = 10, the last one at binomial((3 + 1) + 2, 3) - 1. - _David A. Corneth_, Oct 14 2022

Cf. A000292, A003056, A056557, A056558, A056559, A056560.
See also A194847.
Cf. A088880, A088881, A116608, A325242.

Table[Table[m, {(m+1)(m+2)/2}], {m, 0, 7}] // Flatten (* Jean-François Alcover, Feb 28 2019 *)

a(n)=my(t=polrootsreal(x^3+3*x^2+2*x-6*n)); t[#t]\1 \\ Charles R Greathouse IV, Feb 22 2017

from math import comb
from sympy import integer_nthroot
def A056556(n): return (m:=integer_nthroot(6*(n+1),3)[0])-(nChai Wah Wu, Nov 04 2024

a(n) = floor(x) where x is the (largest real) solution to x^3 + 3x^2 + 2x - 6n = 0; a(A000292(n)) = n+1.
a(n+1) = a(n)+1 if a(n) = A056558(n), otherwise a(n). - Graeme McRae, Jan 09 2007
a(n) = floor(t/3 + 1/t - 1), where t = (81*n + 3*sqrt(729*n^2 - 3))^(1/3). - Ridouane Oudra, Mar 21 2021
a(n) = floor(t + 1/(3*t) - 1), where t = (6*n)^(1/3), for n>=1. - Ridouane Oudra, Nov 04 2022
a(n) = m if n>=binomial(m+2,3) and a(n) = m-1 otherwise where m = floor((6n+6)^(1/3)). - Chai Wah Wu, Nov 04 2024

Incorrect formula deleted by Ridouane Oudra, Nov 04 2022

A056557 Second tetrahedral coordinate.

Original entry on oeis.org

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

Offset: 0

Henry Bottomley, Jun 26 2000

nonn

If {(X,Y,Z)} are triples of nonnegative integers with X >= Y >= Z ordered by X, Y and Z, then X=A056556(n), Y=A056557(n) and Z=A056558(n).

Cf. A000292, A002262, A003056, A056556, A056558, A056559, A056560.

A194847 Write n = C(i,3)+C(j,2)+C(k,1) with i>j>k>=0; sequence gives i values.

Original entry on oeis.org

2, 3, 3, 3, 4, 4, 4, 4, 4, 4, 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, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10

Offset: 0

N. J. A. Sloane, Sep 03 2011

nonn

Each n >= 0 has a unique representation as n = C(i,3)+C(j,2)+C(k,1) with i>j>k>=0. This is the combinatorial number system of degree t = 3, where we get [A194847, A194848, A056558]. For degree t = 2 we get [A002024, A002262] and A138036.

			The i,j,k coordinates for n equal to 0 through 10 are:
0, [2, 1, 0]
1, [3, 1, 0]
2, [3, 2, 0]
3, [3, 2, 1]
4, [4, 1, 0]
5, [4, 2, 0]
6, [4, 2, 1]
7, [4, 3, 0]
8, [4, 3, 1]
9, [4, 3, 2]
10, [5, 1, 0]

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

The [i,j,k] values are [A194847, A194848, A056558], or equivalently [A056556+2, A056557+1, A056558]. See A194849 for the union list of triples.

Cf. also A002024, A002262, A138036.

# Given x and a list a, returns smallest i such that x >= a[i].
whereinlist:=proc(x,a)  local i:
if whattype(a) <> list then ERROR(`a not a list`); fi:
for i from 1 to nops(a) do if x < a[i] then break; fi; od:
RETURN(i-1); end:
t3:=[seq(binomial(n,3),n=0..50)];
t2:=[seq(binomial(n,2),n=0..50)];
t1:=[seq(binomial(n,1),n=0..50)];
for n from 0 to 200 do
i3:=whereinlist(n,t3);
i2:=whereinlist(n-t3[i3],t2);
i1:=whereinlist(n-t3[i3]-t2[i2],t1);
L[n]:=[i3-1,i2-1,i1-1];
od:
[seq(L[n][1],n=0..200)];

from math import comb
from sympy import integer_nthroot
def A194847(n): return (m:=integer_nthroot(6*(n+1),3)[0])+(n>=comb(m+2,3))+1 # Chai Wah Wu, Nov 05 2024

Equals A056556(n) + 2.

A194848 Write n = C(i,3)+C(j,2)+C(k,1) with i>j>k>=0; sequence gives j values.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Sep 03 2011

nonn

See A194847.

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

The [i,j,k] values are [A194847, A194848, A056558].

Maple
```
See A194847.
```

from math import isqrt, comb
from sympy import integer_nthroot
def A194848(n): return (k:=isqrt(r:=n+1-comb((m:=integer_nthroot(6*(n+1),3)[0])-(n(k<<2)*(k+1)+1) # Chai Wah Wu, Nov 04 2024

Equals A056557(n) + 1.

A056559 Tetrahedron with T(t,n,k) = t - n; succession of growing finite triangles with declining values per row.

Original entry on oeis.org

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

Offset: 0

Henry Bottomley, Jun 26 2000

			First triangle: [0]; second triangle: [1; 0 0]; third triangle: [2; 1 1; 0 0 0]; ...

Peter Luschny, Table of n, a(n) for n = 0..10000 (first 105 terms by Henry Bottomley).

Together with A056558 and A056560 might enable reading "by antidiagonals" of cube arrays as 3-dimensional analog of A002262 and A025581 with square arrays.

Bisection (y-coordinates) of A332662.

function a_list(N)
    a = Int[]
    for n in 1:N
        for j in ((k:-1:1) for k in 1:n)
            t = n - j[1]
            for m in j
                push!(a, t)
end end end; a end
A = a_list(10) # Peter Luschny, Feb 19 2020

from math import isqrt, comb
from sympy import integer_nthroot
def A056559(n): return (m:=integer_nthroot(6*(n+1),3)[0])-(a:=nChai Wah Wu, Dec 11 2024

a(n) = A056556(n) - A056557(n).

cp's OEIS Frontend

A127324 Fourth 4-dimensional hyper-tetrahedral coordinate; 4-D analog of A056558.

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A127560 Number of fixed r-celled polyominoes with smallest containing rectangle measuring k by m, read in order r=A056556(n)+1, k=A056560(n)+1, m=A056558(n)+1.

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A002262 Triangle read by rows: T(n,k) = k, 0 <= k <= n, in which row n lists the first n+1 nonnegative integers.

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A014311 Numbers with exactly 3 ones in binary expansion.

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A052217 Numbers whose sum of digits is 3.

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A056556 First tetrahedral coordinate; repeat m (m+1)*(m+2)/2 times.

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