A194884 -id:A194884 - cp's OEIS frontend

A014312 Numbers with exactly 4 ones in binary expansion.

15, 23, 27, 29, 30, 39, 43, 45, 46, 51, 53, 54, 57, 58, 60, 71, 75, 77, 78, 83, 85, 86, 89, 90, 92, 99, 101, 102, 105, 106, 108, 113, 114, 116, 120, 135, 139, 141, 142, 147, 149, 150, 153, 154, 156, 163, 165, 166, 169, 170, 172, 177, 178, 180, 184, 195, 197

Offset: 1

Al Black (gblack(AT)nol.net)

T. D. Noe and Ivan Neretin, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
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.

Cf. A090706.
Cf. A000079, A018900, A014311, A014313, A023688, A023689, A023690, A023691 (Hamming weight = 1, 2, ..., 9), A057168.
Cf. A194882, A194883, A194884, A127324.

Select[ Range[ 180 ], (Count[ IntegerDigits[ #, 2 ], 1 ]==4)& ] (* Olivier Gérard *)

for(n=0,10^3,if(hammingweight(n)==4,print1(n,", "))); \\ Joerg Arndt, Mar 04 2014

print1(t=15); for(i=2, 50, print1(", "t=A057168(t))) \\ M. F. Hasler, Aug 27 2014

$N = 4;
my $vector = 2 ** $N - 1;  # first key (15)
for (1..100) {
  print "$vector, ";
  my ($v, $d) = ($vector, 0);
  until ($v & 1 or !$v) { $d = ($d << 1)|1; $v >>= 1 }
  $vector += $d + 1 + (($v ^ ($v + 1)) >> 2);  # next key
} # Ruud H.G. van Tol, Mar 02 2014

A014312_list = [2**a+2**b+2**c+2**d for a in range(3,6) for b in range(2,a) for c in range(1,b) for d in range(c)] # Chai Wah Wu, Jan 24 2021

from itertools import islice
def A014312_gen(): # generator of terms
    yield (n:=15)
    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)
A014312_list = list(islice(A014312_gen(),20)) # Chai Wah Wu, Mar 10 2025

pub const fn next_choice(value: usize) -> usize {
  // Passing a term will return the next number in the sequence
  let zeros = value.trailing_zeros();
  let ones = (value >> zeros).trailing_ones();
  value + (1 << zeros) + (1 << (ones - 1)) - 1
} // Andrew Bennett, Jan 07 2022

a(n+1) = A057168(a(n)). - M. F. Hasler, Aug 27 2014
a(n) = 2^A194882(n-1) + 2^A194883(n-1) + 2^A194884(n-1) + 2^A127324(n-1). - Ridouane Oudra, Sep 06 2020
Sum_{n>=1} 1/a(n) = 1.399770961748474333075618147113153558623203796657745865012742162098738541849... (calculated using Baillie's irwinSums.m, see Links). - Amiram Eldar, Feb 14 2022

Extension by Olivier Gérard

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

Original entry on oeis.org

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

A194882 Write n = C(i,4)+C(j,3)+C(k,2)+C(l,1) with i>j>k>l>=0; sequence gives i values.

Original entry on oeis.org

3, 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, 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, 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, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8

Offset: 0

N. J. A. Sloane, Sep 04 2011

nonn

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]. For degree t = 3 see A194847.

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

Equals A127321 + 3.

Cf. A194882-A194884, A127324, A194885; A194847, A194848, A056558, A194849.

from math import comb
from sympy import integer_nthroot
def A194882(n): return (m:=integer_nthroot(24*(n+2),4)[0]+1)+(n>=comb(m+1,4)) # Chai Wah Wu, Dec 10 2024

a(n) = m if n < binomial(m+1,4) and a(n) = m+1 otherwise where m = 1+floor((24*(n+2))^(1/4)). - Chai Wah Wu, Dec 10 2024

A194883 Write n = C(i,4)+C(j,3)+C(k,2)+C(l,1) with i>j>k>l>=0; sequence gives j values.

Original entry on oeis.org

2, 2, 3, 3, 3, 2, 3, 3, 3, 4, 4, 4, 4, 4, 4, 2, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 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, 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

Offset: 0

N. J. A. Sloane, Sep 04 2011

nonn

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]. For degree t = 3 see A194847.

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

Equals A127322 + 2.

Cf. A194882-A194884, A127324, A194885; A194847, A194848, A056558, A194849.

A194885 Write n = C(i,4)+C(j,3)+C(k,2)+C(l,1) with i>j>k>l>=0; let L[n] = [i,j,k,l]; sequence gives list of quadruples L[n], n >= 0.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Sep 04 2011

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]. For degree t = 3 see A194847.

			List of quadruples begins:
[3, 2, 1, 0]
[4, 2, 1, 0]
[4, 3, 1, 0]
[4, 3, 2, 0]
[4, 3, 2, 1]
[5, 2, 1, 0]
[5, 3, 1, 0]
[5, 3, 2, 0]
[5, 3, 2, 1]
[5, 4, 1, 0]
[5, 4, 2, 0]
...

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

The four columns are [A194882, A194883, A194885, A127324], or equivalently [A127321+3, A127322+2, A127323+1, A127324].

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A014312 Numbers with exactly 4 ones in binary expansion.

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

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A194882 Write n = C(i,4)+C(j,3)+C(k,2)+C(l,1) with i>j>k>l>=0; sequence gives i values.

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A194883 Write n = C(i,4)+C(j,3)+C(k,2)+C(l,1) with i>j>k>l>=0; sequence gives j values.

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A194885 Write n = C(i,4)+C(j,3)+C(k,2)+C(l,1) with i>j>k>l>=0; let L[n] = [i,j,k,l]; sequence gives list of quadruples L[n], n >= 0.

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