A056556 -id:A056556 - cp's OEIS frontend

A127323 Third 4-dimensional hyper-tetrahedral coordinate; 4-D analog of A056557.

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

Offset: 0

Graeme McRae, Jan 10 2007

nonn

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.

			a(23)=2 because a(A000332(2+3)+A000292(2)+A000217(2)) = a(A000332(2+3)+A000292(2)+A000217(2+1)-1) = 2, so a(22) = a(24) = 2.
See A127321 for a table of A127321, A127322, A127323, A127324.

Cf. A127321, A127322, A127324, A056556, A056557, A056558, A000332, A000292, A000217.

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

A331195 Three-column table read by rows: triples (i,j,k) in order sorted from the left.

Original entry on oeis.org

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

Offset: 0

Mehmet A. Ates, Jun 08 2020

			For n=[0,1,2] to n=[12,13,14], a[n,n+1,n+2] counts up as such: [0,0,0], [1,0,0], [1,1,0], [1,1,1], [2,0,0], etc.

Mehmet A. Ates, Table of n, a(n) for n = 0..10000

Cf. A056556, A056557, A056558, A330709.

See A372667 for the norms of these triples.

ThreeDVectors = List[];
SeqSize = 10;
For[i = 0, i <= SeqSize, i++,
  For[j = 0, j <= i, j++,
    For[k = 0, k <= j, k++,
      AppendTo[ThreeDVectors, {i, j, k}]
    ]
  ]
];
Flatten[ThreeDVectors]

from math import comb, isqrt
from sympy import integer_nthroot
def A331195(n): return (m:=integer_nthroot((n<<1)+6,3)[0])-(n<3*comb(m+2,3)) if not (a:=n%3) else (k:=isqrt(r:=(b:=n//3)+1-comb((m:=integer_nthroot((n<<1)-1,3)[0])-(b=comb(m+2,3))+1,3))-comb((k:=isqrt(m:=r+1<<1))+(m>k*(k+1)),2) # Chai Wah Wu, Nov 23 2024

a(3*n) = A056556(n).
a(3*n+1) = A056557(n).
a(3*n+2) = A056558(n).

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

A070770 b + c + d where b >= c >= d >= 0 ordered by b then c then d.

Original entry on oeis.org

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

Offset: 0

Henry Bottomley, May 06 2002

			Triangle begins:
  0,
  ;
  1;
  2, 3;
  ;
  2;
  3, 4;
  4, 5, 6;
  ;
  3;
  4, 5,
  5, 6, 7;
  6, 7, 8, 9;
  ;
  4;
  5, 6;
  6, 7,  8;
  7, 8,  9, 10;
  8, 9, 10, 11, 12;
  ;
  ...

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A001477, A051162, A070771, A070772 for similar sequences with different numbers of terms summed.

seq(seq(seq(b+c+d,d=0..c),c=0..b),b=0..10); # Robert Israel, Jun 21 2018

for(x=0,5,for(y=0,x,for(z=0,y,print1(x+y+z", ")))) \\ Charles R Greathouse IV, Sep 17 2015

from math import isqrt, comb
from sympy import integer_nthroot
def A070770(n): return (m:=integer_nthroot(6*(n+1),3)[0])+(a:=n>=comb(m+2,3))+(k:=isqrt(b:=(c:=n+1-comb(m+a+1,3))<<1))-((b<<2)<=(k<<2)*(k+1)+1)+c-2-comb(k+(b>k*(k+1)),2) # Chai Wah Wu, Dec 11 2024

a(n) = A056556(n) + A056557(n) + A056558(n).

A357773 Odd numbers with two zeros in their binary expansion.

Original entry on oeis.org

9, 19, 21, 25, 39, 43, 45, 51, 53, 57, 79, 87, 91, 93, 103, 107, 109, 115, 117, 121, 159, 175, 183, 187, 189, 207, 215, 219, 221, 231, 235, 237, 243, 245, 249, 319, 351, 367, 375, 379, 381, 415, 431, 439, 443, 445, 463, 471, 475, 477, 487, 491, 493, 499, 501

Offset: 1

Bernard Schott, Oct 12 2022

A048490 \ {1} is a subsequence, since for m >= 1, A048490(m) = 8*2^m - 7 has 11..11001 with m starting 1 for binary expansion.
A153894 \ {4} is a subsequence, since for m >= 1, A153894(m) = 5*2^m - 1 has 10011..11 with m trailing 1 for binary expansion.
A220236 is a subsequence, since for m >= 1, A220236(m) = 2^(2*m + 2) - 2^(m + 1) - 2^m - 1 has 11..110011..11 with m starting 1 and m trailing 1 for binary expansion.
For k > 2, there are (k-1)*(k-2)/2 terms between 2^k and 2^(k+1), or equivalently (k-1)*(k-2)/2 terms with k+1 bits.
Binary expansion of a(n) is A357774(n).
{4*a(n), n>0} form a subsequence of A353654 (numbers with two trailing 0 bits and two other 0 bits).

Cf. A018900, A005408, A023416, A353654, A357774.
Odd numbers with k zeros in their binary expansion: A000225 (k=0), A190620 (k=1).
Subsequences: A048490 \ {1}, A153894 \ {4}, A220236.

seq(seq(seq(2^n-1-2^i-2^j,j=i-1..1,-1),i=n-2..1,-1),n=4..10); # Robert Israel, Oct 13 2022

Select[Range[1, 500, 2], DigitCount[#, 2, 0] == 2 &] (* Amiram Eldar, Oct 12 2022 *)

isok(k) = (k%2) && (#binary(k) == hammingweight(k)+2); \\ Michel Marcus, Oct 13 2022

list(lim)=my(v=List()); for(n=4,logint(lim\=1,2)+1, my(N=2^n-1); forstep(a=n-2,2,-1, my(A=N-1<lim, break(2)); listput(v,t)))); Vec(v) \\ Charles R Greathouse IV, Oct 21 2022

def a(n):
    m = 0
    while m*(m+1)*(m+2)//6 <= n: m += 1
    m -= 1 # m = A056556(n-1)
    k, r, j = m + 4, n - m*(m+1)*(m+2)//6, 0
    while r >= 0: r -= (m+1-j); j += 1
    j += 1
    return 2**k - 2**(k-j) - 2**(-r) - 1
print([a(n) for n in range(60)]) # Michael S. Branicky, Oct 12 2022

# faster version for generating initial segment of sequence
from itertools import combinations, count, islice
def agen():
    for d in count(4):
        b, c = 2**d - 1, 2**(d-1)
        for i, j in combinations(range(1, d-1), 2):
            yield b - (c >> i) - (c >> j)
print(list(islice(agen(), 60))) # Michael S. Branicky, Oct 13 2022

from math import comb, isqrt
from sympy import integer_nthroot
def A357773(n):
    a = (m:=integer_nthroot(6*n, 3)[0])+(n>comb(m+2,3))+3
    b = isqrt((j:=comb(a-1,3)-n+1)<<3)+3>>1
    c = j-comb((r:=isqrt(w:=j<<1))+(w>r*(r+1)),2)
    return (1<Chai Wah Wu, Dec 17 2024

A023416(a(n)) = 2.

a((n-1)*(n-2)*(n-3)/6 - (i-1)*(i-2)/2 - (j-1)) = 2^n - 2^i - 2^j - 1 for 1 <= j < i <= n-2. - Robert Israel, Oct 13 2022

a(11) and beyond from Michael S. Branicky, Oct 12 2022

A007665 Tower of Hanoi with 5 pegs.

Original entry on oeis.org

1, 3, 5, 7, 11, 15, 19, 23, 27, 31, 39, 47, 55, 63, 71, 79, 87, 95, 103, 111, 127, 143, 159, 175, 191, 207, 223, 239, 255, 271, 287, 303, 319, 335, 351, 383, 415, 447, 479, 511, 543, 575, 607, 639, 671, 703, 735, 767, 799

Offset: 1

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

nonn

A. Brousseau, Tower of Hanoi with more pegs, J. Recreational Math., 8 (1975-1976), 169-176.
Cull, Paul; Ecklund, E. F. On the Towers of Hanoi and generalized Towers of Hanoi problems. Proceedings of the thirteenth Southeastern conference on combinatorics, graph theory and computing (Boca Raton, Fla., 1982). Congr. Numer. 35 (1982), 229--238. MR0725883(85a:68059). - N. J. A. Sloane, Apr 08 2012
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
D. Wood, Towers of Brahma and Hanoi revisited, J. Recreational Math., 14 (1981), 17-24.

S. Alejandre, Legend of Towers of Hanoi
J.-P. Allouche, Note on the cyclic towers of Hanoi, Theoret. Comput. Sci., 123 (1994), 3-7.
A. Brousseau, Tower of Hanoi with more pegs, J. Recreational Math 8.3 (1975-6), 169-176. (Annotated scanned copy)
Eric Weisstein's World of Mathematics, Towers of Hanoi
Index entries for sequences related to Towers of Hanoi

Cf. A007664. A182058, A056556.

terms = 100;
A056556 = Table[Table[m, {(m+1)(m+2)/2}], {m, 0, (6 terms)^(1/3) // Ceiling}] // Flatten;
a[n_] := With[{t = A056556[[n+1]]}, -1+(1+t(t-1)/2+n-t(t+1)(t+2)/6)*2^t];
Array[a, terms] (* Jean-François Alcover, Feb 28 2019 *)

PARI
```
m=1;n=1;while(nK. Spage, Oct 23 2009
```

a(n) = - 1 + (1 + A056556(n)*(A056556(n) - 1)/2 + n - A056556(n)*(A056556(n) + 1)*(A056556(n) + 2)/6)*2^A056556(n). - Daniele Parisse, Feb 06 2001

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.

Original entry on oeis.org

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.

A056553 Smallest 4th-power divisible by n divided by largest 4th-power which divides n.

Original entry on oeis.org

1, 16, 81, 16, 625, 1296, 2401, 16, 81, 10000, 14641, 1296, 28561, 38416, 50625, 1, 83521, 1296, 130321, 10000, 194481, 234256, 279841, 1296, 625, 456976, 81, 38416, 707281, 810000, 923521, 16, 1185921, 1336336, 1500625, 1296, 1874161, 2085136

Offset: 1

Henry Bottomley, Jun 25 2000

			a(64) = 16 because smallest 4th power divisible by 64 is 256 and largest 4th power which divides 64 is 16 and 256/16 = 16.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Henry Bottomley, Some Smarandache-type multiplicative sequences

Cf. A000190, A008835, A053164, A053165, A053166, A053167, A056551, A056554, A056555.

Cf. A013662, A013678.

f[p_, e_] := p^If[Divisible[e, 4], 0, 1]; a[n_] := (Times @@ (f @@@ FactorInteger[ n]))^4; Array[a, 100] (* Amiram Eldar, Aug 29 2019*)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2]%4, f[i,1], 1))^4; } \\ Amiram Eldar, Oct 27 2022

a(n) = A053167(n)/A008835(n) = A056556(n)*A053165(n) = A056554(n)^4.
From Amiram Eldar, Oct 27 2022: (Start)
Multiplicative with a(p^e) = 1 if e is divisible by 4, and a(p^e) = p^4 otherwise.
Sum_{k=1..n} a(k) ~ c * n^5, where c = (zeta(20)/(5*zeta(4))) * Product_{p prime} (1 - 1/p^2 + 1/p^4 - 1/p^7 + 1/p^8) = 0.123026157003... . (End)

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.

A023660 Convolution of odd numbers and A023533.

Original entry on oeis.org

1, 3, 5, 8, 12, 16, 20, 24, 28, 33, 39, 45, 51, 57, 63, 69, 75, 81, 87, 94, 102, 110, 118, 126, 134, 142, 150, 158, 166, 174, 182, 190, 198, 206, 215, 225, 235, 245, 255, 265, 275, 285, 295, 305, 315, 325, 335, 345, 355, 365, 375, 385, 395, 405

Offset: 1

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A005408, A023533, A023543, A056556.

A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
[(&+[(2*k+1)*A023533(n-k): k in [0..n-1]]): n in [1..80]]; // G. C. Greubel, Jul 17 2022

Table[(2*k+1)*n + 6*Binomial[n+2,4], {n, 7}, {k,0,n*(n+3)/2}]//Flatten (* G. C. Greubel, Jul 17 2022 *)

def A023660(n, k): return (2*k+1)*n + 6*binomial(n+2, 4)
flatten([[A023660(n,k) for k in (0..n*(n+3)/2)] for n in (1..7)]) # G. C. Greubel, Jul 17 2022

From G. C. Greubel, Jul 17 2022: (Start)
a(n) = Sum_{j=0..n-1} (2*j+1)*A023533(n-j).
a(n) = 2*A023543(n-1) + A056556(n).
T(n, k) = (2*k+1)*n + 6*binomial(n+2, 4), for 0 <= k <= n*(n+3)/2 and n >= 1 (as an irregular triangle). (End)

cp's OEIS Frontend

A127323 Third 4-dimensional hyper-tetrahedral coordinate; 4-D analog of A056557.

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A331195 Three-column table read by rows: triples (i,j,k) in order sorted from the left.

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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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A070770 b + c + d where b >= c >= d >= 0 ordered by b then c then d.

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Maple

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A357773 Odd numbers with two zeros in their binary expansion.

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A007665 Tower of Hanoi with 5 pegs.

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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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A056553 Smallest 4th-power divisible by n divided by largest 4th-power which divides n.

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