cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 16 results. Next

A085359 Third self-convolution of A085357 (residues of binomial(3k,k)/(2k+1) modulo 2).

Original entry on oeis.org

1, 3, 6, 7, 9, 12, 16, 15, 15, 21, 30, 30, 28, 30, 36, 31, 27, 33, 48, 51, 51, 54, 66, 60, 52, 60, 78, 76, 66, 66, 76, 63, 51, 57, 78, 81, 81, 90, 114, 105, 93, 111, 144, 138, 120, 126, 150, 126, 100, 108, 144, 148, 138, 138, 166, 150, 126, 138, 174, 168, 142, 138, 156
Offset: 0

Views

Author

Paul D. Hanna, Jun 25 2003

Keywords

Comments

a(n) - A085357(n-1) is always even for n > 0.

Crossrefs

Cf. A001764 (ternary trees), A085357.

Formula

G.f.: 1 + x*A(x) = A085357(x)^3 (mod 2), where A085357(x) = Sum_{k>=0} A085357(k)*x^k.

A156552 Unary-encoded compressed factorization of natural numbers.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 8, 7, 6, 9, 16, 11, 32, 17, 10, 15, 64, 13, 128, 19, 18, 33, 256, 23, 12, 65, 14, 35, 512, 21, 1024, 31, 34, 129, 20, 27, 2048, 257, 66, 39, 4096, 37, 8192, 67, 22, 513, 16384, 47, 24, 25, 130, 131, 32768, 29, 36, 71, 258, 1025, 65536, 43, 131072, 2049, 38, 63, 68, 69, 262144
Offset: 1

Views

Author

Leonid Broukhis, Feb 09 2009

Keywords

Comments

The primes become the powers of 2 (2 -> 1, 3 -> 2, 5 -> 4, 7 -> 8); the composite numbers are formed by taking the values for the factors in the increasing order, multiplying them by the consecutive powers of 2, and summing. See the Example section.
From Antti Karttunen, Jun 27 2014: (Start)
The odd bisection (containing even terms) halved gives A244153.
The even bisection (containing odd terms), when one is subtracted from each and halved, gives this sequence back.
(End)
Question: Are there any other solutions that would satisfy the recurrence r(1) = 0; and for n > 1, r(n) = Sum_{d|n, d>1} 2^A033265(r(d)), apart from simple variants 2^k * A156552(n)? See also A297112, A297113. - Antti Karttunen, Dec 30 2017

Examples

			For 84 = 2*2*3*7 -> 1*1 + 1*2 + 2*4 + 8*8 =  75.
For 105 = 3*5*7 -> 2*1 + 4*2 + 8*4 = 42.
For 137 = p_33 -> 2^32 = 4294967296.
For 420 = 2*2*3*5*7 -> 1*1 + 1*2 + 2*4 + 4*8 + 8*16 = 171.
For 147 = 3*7*7 = p_2 * p_4 * p_4 -> 2*1 + 8*2 + 8*4 = 50.

Links

Crossrefs

One less than A005941.
Inverse permutation: A005940 with starting offset 0 instead of 1.
Cf. A000079, A000120, A001222, A052126, A054429, A061395, A064216, A064989, A003188, A243071, A243065-A243066, A244153, A243354, A112798, A125106, A056239, A161511.
Cf. also A297106, A297112 (Möbius transform), A297113, A153013, A290308, A300827, A323243, A323244, A323247, A324201, A324812 (n for which a(n) is a square), A324813, A324822, A324823, A324398, A324713, A324815, A324819, A324865, A324866, A324867.
Other related permutations: A253551, A253792, A253564, A253791, A277195, A297163, A297164, A297165, A297166, A302023, A305418, A322863, A322864.

Programs

  • Mathematica
    Table[Floor@ Total@ Flatten@ MapIndexed[#1 2^(#2 - 1) &, Flatten[ Table[2^(PrimePi@ #1 - 1), {#2}] & @@@ FactorInteger@ n]], {n, 67}] (* Michael De Vlieger, Sep 08 2016 *)
  • PARI
    a(n) = {my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ David A. Corneth, Mar 08 2019
  • PARI
    A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n)))); \\ (based on the given recurrence) - Antti Karttunen, Mar 08 2019
  • Perl
    # Program corrected per instructions from Leonid Broukhis. - Antti Karttunen, Jun 26 2014
# However, it gives correct answers only up to n=136, before corruption by a wrap-around effect.
# Note that the correct answer for n=137 is A156552(137) = 4294967296.
$max = $ARGV[0];
$pow = 0;
foreach $i (2..$max) {
@a = split(/ /, `factor $i`);
shift @a;
$shift = 0;
$cur = 0;
while ($n = int shift @a) {
$prime{$n} = 1 << $pow++ if !defined($prime{$n});
$cur |= $prime{$n} << $shift++;
}
print "$cur, ";
}
print "\n";
(Scheme, with memoization-macro definec from Antti Karttunen's IntSeq-library, two different implementations)
(definec (A156552 n) (cond ((= n 1) 0) (else (+ (A000079 (+ -2 (A001222 n) (A061395 n))) (A156552 (A052126 n))))))
(definec (A156552 n) (cond ((= 1 n) (- n 1)) ((even? n) (+ 1 (* 2 (A156552 (/ n 2))))) (else (* 2 (A156552 (A064989 n))))))
;; Antti Karttunen, Jun 26 2014
  • Python
    from sympy import primepi, factorint
def A156552(n): return sum((1<Chai Wah Wu, Mar 10 2023

Formula

From Antti Karttunen, Jun 26 2014: (Start)
a(1) = 0, a(n) = A000079(A001222(n)+A061395(n)-2) + a(A052126(n)).
a(1) = 0, a(2n) = 1+2*a(n), a(2n+1) = 2*a(A064989(2n+1)). [Compare to the entanglement recurrence A243071].
For n >= 0, a(2n+1) = 2*A244153(n+1). [Follows from the latter clause of the above formula.]
a(n) = A005941(n) - 1.
As a composition of related permutations:
a(n) = A003188(A243354(n)).
a(n) = A054429(A243071(n)).
For all n >= 1, A005940(1+a(n)) = n and for all n >= 0, a(A005940(n+1)) = n. [The offset-0 version of A005940 works as an inverse for this permutation.]
This permutations also maps between the partition-lists A112798 and A125106:
A056239(n) = A161511(a(n)). [The sums of parts of each partition (the total sizes).]
A003963(n) = A243499(a(n)). [And also the products of those parts.]
(End)
From Antti Karttunen, Oct 09 2016: (Start)
A161511(a(n)) = A056239(n).
A029837(1+a(n)) = A252464(n). [Binary width of terms.]
A080791(a(n)) = A252735(n). [Number of nonleading 0-bits.]
A000120(a(n)) = A001222(n). [Binary weight.]
For all n >= 2, A001511(a(n)) = A055396(n).
For all n >= 2, A000120(a(n))-1 = A252736(n). [Binary weight minus one.]
A252750(a(n)) = A252748(n).
a(A250246(n)) = A252754(n).
a(A005117(n)) = A277010(n). [Maps squarefree numbers to a permutation of A003714, fibbinary numbers.]
A085357(a(n)) = A008966(n). [Ditto for their characteristic functions.]
For all n >= 0:
a(A276076(n)) = A277012(n).
a(A276086(n)) = A277022(n).
a(A260443(n)) = A277020(n).
(End)
From Antti Karttunen, Dec 30 2017: (Start)
For n > 1, a(n) = Sum_{d|n, d>1} 2^A033265(a(d)). [See comments.]
More linking formulas:
A106737(a(n)) = A000005(n).
A290077(a(n)) = A000010(n).
A069010(a(n)) = A001221(n).
A136277(a(n)) = A181591(n).
A132971(a(n)) = A008683(n).
A106400(a(n)) = A008836(n).
A268411(a(n)) = A092248(n).
A037011(a(n)) = A010052(n) [conjectured, depends on the exact definition of A037011].
A278161(a(n)) = A046951(n).
A001316(a(n)) = A061142(n).
A277561(a(n)) = A034444(n).
A286575(a(n)) = A037445(n).
A246029(a(n)) = A181819(n).
A278159(a(n)) = A124859(n).
A246660(a(n)) = A112624(n).
A246596(a(n)) = A069739(n).
A295896(a(n)) = A053866(n).
A295875(a(n)) = A295297(n).
A284569(a(n)) = A072411(n).
A286574(a(n)) = A064547(n).
A048735(a(n)) = A292380(n).
A292272(a(n)) = A292382(n).
A244154(a(n)) = A048673(n), a(A064216(n)) = A244153(n).
A279344(a(n)) = A279339(n), a(A279338(n)) = A279343(n).
a(A277324(n)) = A277189(n).
A037800(a(n)) = A297155(n).
For n > 1, A033265(a(n)) = 1+A297113(n).
(End)
From Antti Karttunen, Mar 08 2019: (Start)
a(n) = A048675(n) + A323905(n).
a(A324201(n)) = A000396(n), provided there are no odd perfect numbers.
The following sequences are derived from or related to the base-2 expansion of a(n):
A000265(a(n)) = A322993(n).
A002487(a(n)) = A323902(n).
A005187(a(n)) = A323247(n).
A324288(a(n)) = A324116(n).
A323505(a(n)) = A323508(n).
A079559(a(n)) = A323512(n).
A085405(a(n)) = A323239(n).
The following sequences are obtained by applying to a(n) a function that depends on the prime factorization of its argument, which goes "against the grain" because a(n) is the binary code of the factorization of n, which in these cases is then factored again:
A000203(a(n)) = A323243(n).
A033879(a(n)) = A323244(n) = 2*a(n) - A323243(n),
A294898(a(n)) = A323248(n).
A000005(a(n)) = A324105(n).
A000010(a(n)) = A324104(n).
A083254(a(n)) = A324103(n).
A001227(a(n)) = A324117(n).
A000593(a(n)) = A324118(n).
A001221(a(n)) = A324119(n).
A009194(a(n)) = A324396(n).
A318458(a(n)) = A324398(n).
A192895(a(n)) = A324100(n).
A106315(a(n)) = A324051(n).
A010052(a(n)) = A324822(n).
A053866(a(n)) = A324823(n).
A001065(a(n)) = A324865(n) = A323243(n) - a(n),
A318456(a(n)) = A324866(n) = A324865(n) OR a(n),
A318457(a(n)) = A324867(n) = A324865(n) XOR a(n),
A318458(a(n)) = A324398(n) = A324865(n) AND a(n),
A318466(a(n)) = A324819(n) = A323243(n) OR 2*a(n),
A318467(a(n)) = A324713(n) = A323243(n) XOR 2*a(n),
A318468(a(n)) = A324815(n) = A323243(n) AND 2*a(n).
(End)

Extensions

More terms from Antti Karttunen, Jun 28 2014

A008966 a(n) = 1 if n is squarefree, otherwise 0.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3, 1).
The infinite lower triangular matrix with A008966 on the main diagonal and the rest zeros is the square of triangle A143255. - Gary W. Adamson, Aug 02 2008

Links

Crossrefs

Cf. A005117, A008836 (Dirichlet inverse), A013928 (partial sums).
Cf. A179211, A179212, A179213, A179214, A179215.
Cf. A020639, A073576, A087188.
Cf. A124010, A212793, A085357, A156552.
Parity of A002033.
Cf. A047999, A036987
Cf. A082020 (Dgf at s=2), A157289 (Dgf at s=3), A157290 (Dgf at s=4).

Programs

  • Haskell
    a008966 = abs . a008683
-- Reinhard Zumkeller, Dec 13 2015, Dec 15 2014, May 27 2012, Jan 25 2012
  • Magma
    [ Abs(MoebiusMu(n)) : n in [1..100]];
  • Maple
    A008966 := proc(n) if numtheory[issqrfree](n) then 1 ; else 0 ; end if; end proc: # R. J. Mathar, Mar 14 2011
  • Mathematica
    A008966[n_] := Abs[MoebiusMu[n]]; Table[A008966[n], {n, 100}] (* Enrique Pérez Herrero, Apr 15 2010 *)
Table[If[SquareFreeQ[n],1,0],{n,100}] (* or *) Boole[SquareFreeQ/@ Range[ 100]] (* Harvey P. Dale, Feb 28 2015 *)
  • MuPAD
    func(abs(numlib::moebius(n)), n):
  • PARI
    a(n)=if(n<1,0,direuler(p=2,n,1+X))[n]
  • PARI
    a(n)=issquarefree(n) \\ Michel Marcus, Feb 22 2015
  • Python
    from sympy import factorint
def A008966(n): return int(max(factorint(n).values(),default=1)==1) # Chai Wah Wu, Apr 05 2023

Formula

Dirichlet g.f.: zeta(s)/zeta(2s).
a(n) = abs(mu(n)), where mu is the Moebius function (A008683).
a(n) = 0^(bigomega(n) - omega(n)), where bigomega(n) and omega(n) are the numbers of prime factors of n with and without repetition (A001222, A001221, A046660). - Reinhard Zumkeller, Apr 05 2003
Multiplicative with p^e -> 0^(e - 1), p prime and e > 0. - Reinhard Zumkeller, Jul 15 2003
a(n) = 0^(A046951(n) - 1). - Reinhard Zumkeller, May 20 2007
a(n) = 1 - A107078(n). - Reinhard Zumkeller, Oct 03 2008
a(n) = floor(rad(n)/n), where rad() is A007947. - Enrique Pérez Herrero, Nov 13 2009
A175046(n) = a(n)*A073311(n). - Reinhard Zumkeller, Apr 05 2010
a(n) = floor(A000005(n^2)/A007425(n)). - Enrique Pérez Herrero, Apr 15 2010
a(A005117(n)) = 1; a(A013929(n)) = 0; a(n) = A013928(n + 1) - A013928(n). - Reinhard Zumkeller, Jul 05 2010
a(n) * A112526(n) = A063524(n). - Reinhard Zumkeller, Sep 16 2011
a(n) = mu(n) * lambda(n) = A008836(n) * A008683(n). - Enrique Pérez Herrero, Nov 29 2013
a(n) = Sum_{d|n} 2^omega(d)*mu(n/d). - Geoffrey Critzer, Feb 22 2015
a(n) = A085357(A156552(n)). - Antti Karttunen, Mar 06 2017
Limit_{n->oo} (1/n)*Sum_{j=1..n} a(j) = 6/Pi^2. - Andres Cicuttin, Aug 13 2017
a(1) = 1; a(n) = -Sum_{d|n, d < n} (-1)^bigomega(n/d) * a(d). - Ilya Gutkovskiy, Mar 10 2021

Extensions

Deleted an unclear comment. - N. J. A. Sloane, May 30 2021

A003714 Fibbinary numbers: if n = F(i1) + F(i2) + ... + F(ik) is the Zeckendorf representation of n (i.e., write n in Fibonacci number system) then a(n) = 2^(i1 - 2) + 2^(i2 - 2) + ... + 2^(ik - 2). Also numbers whose binary representation contains no two adjacent 1's.

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 9, 10, 16, 17, 18, 20, 21, 32, 33, 34, 36, 37, 40, 41, 42, 64, 65, 66, 68, 69, 72, 73, 74, 80, 81, 82, 84, 85, 128, 129, 130, 132, 133, 136, 137, 138, 144, 145, 146, 148, 149, 160, 161, 162, 164, 165, 168, 169, 170, 256, 257, 258, 260, 261, 264
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

The name "Fibbinary" is due to Marc LeBrun.
"... integers whose binary representation contains no consecutive ones and noticed that the number of such numbers with n bits was fibonacci(n)". [posting to sci.math by Bob Jenkins (bob_jenkins(AT)burtleburtle.net), Jul 17 2002]
From Benoit Cloitre, Mar 08 2003: (Start)
A number m is in the sequence if and only if C(3m, m) (or equally, C(3m, 2m)) is odd.
a(n) == A003849(n) (mod 2). (End)
Numbers m such that m XOR 2*m = 3*m. - Reinhard Zumkeller, May 03 2005. [This implies that A003188(2*a(n)) = 3*a(n) holds for all n.]
Numbers whose base-2 representation contains no two adjacent ones. For example, m = 17 = 10001_2 belongs to the sequence, but m = 19 = 10011_2 does not. - Ctibor O. Zizka, May 13 2008
m is in the sequence if and only if the central Stirling number of the second kind S(2*m, m) = A007820(m) is odd. - O-Yeat Chan (math(AT)oyeat.com), Sep 03 2009
A000120(3*a(n)) = 2*A000120(a(n)); A002450 is a subsequence.
Every nonnegative integer can be expressed as the sum of two terms of this sequence. - Franklin T. Adams-Watters, Jun 11 2011
Subsequence of A213526. - Arkadiusz Wesolowski, Jun 20 2012
This is also the union of A215024 and A215025 - see the Comment in A014417. - N. J. A. Sloane, Aug 10 2012
The binary representation of each term m contains no two adjacent 1's, so we have (m XOR 2m XOR 3m) = 0, and thus a two-player Nim game with three heaps of (m, 2m, 3m) stones is a losing configuration for the first player. - V. Raman, Sep 17 2012
Positions of zeros in A014081. - John Keith, Mar 07 2022
These numbers are similar to Fibternary numbers A003726, Tribbinary numbers A060140 and Tribternary numbers. This sequence is a subsequence of Fibternary numbers A003726. The number of Fibbinary numbers less than any power of two is a Fibonacci number. We can generate this sequence recursively: start with 0 and 1; then, if x is in the sequence add 2x and 4x+1 to the sequence. The Fibbinary numbers have the property that the n-th Fibbinary number is even if the n-th term of the Fibonacci word is a. Respectively, the n-th Fibbinary number is odd (of the form 4x+1) if the n-th term of the Fibonacci word is b. Every number has a Fibbinary multiple. - Tanya Khovanova and PRIMES STEP Senior, Aug 30 2022
This is the ordered set S of numbers defined recursively by: 0 is in S; if x is in S, then 2*x and 4*x + 1 are in S. See Kimberling (2006) Example 3, in references below. - Harry Richman, Jan 31 2024

Examples

			From _Joerg Arndt_, Jun 11 2011: (Start)
In the following, dots are used for zeros in the binary representation:
  a(n)  binary(a(n))  n
    0:    .......     0
    1:    ......1     1
    2:    .....1.     2
    4:    ....1..     3
    5:    ....1.1     4
    8:    ...1...     5
    9:    ...1..1     6
   10:    ...1.1.     7
   16:    ..1....     8
   17:    ..1...1     9
   18:    ..1..1.    10
   20:    ..1.1..    11
   21:    ..1.1.1    12
   32:    .1.....    13
   33:    .1....1    14
   34:    .1...1.    15
   36:    .1..1..    16
   37:    .1..1.1    17
   40:    .1.1...    18
   41:    .1.1..1    19
   42:    .1.1.1.    20
   64:    1......    21
   65:    1.....1    22
(End)

References

  • Donald E. Knuth, The Art of Computer Programming: Fundamental Algorithms, Vol. 1, 2nd ed., Addison-Wesley, 1973, pp. 85, 493.

Links

Crossrefs

A007088(a(n)) = A014417(n) (same sequence in binary). Complement: A004780. Char. function: A085357. Even terms: A022340, odd terms: A022341. First difference: A129761.
Other sequences based on similar restrictions on binary expansion: A003726 & A278038, A003754, A048715, A048718, A107907, A107909.
Cf. A000045, A005203, A005590, A007895, A037011, A048728, A048679, A056017, A060112, A072649, A083368, A089939, A106027, A106028, A116361.
3*a(n) is in A001969.
Cf. also A215022-A215025, A242407.
Cf. A014081 (count 11 bits).

Programs

  • Haskell
    import Data.Set (Set, singleton, insert, deleteFindMin)
a003714 n = a003714_list !! n
a003714_list = 0 : f (singleton 1) where
   f :: Set Integer -> [Integer]
   f s = m : (f $ insert (4*m + 1) $ insert (2*m) s')
         where (m, s') = deleteFindMin s
-- Reinhard Zumkeller, Jun 03 2012, Feb 07 2012
  • Maple
    A003714 := proc(n)
    option remember;
    if n < 3 then
        n ;
    else
        2^(A072649(n)-1) + procname(n-combinat[fibonacci](1+A072649(n))) ;
    end if;
end proc:
seq(A003714(n),n=0..10) ;
# To produce a table giving n, a(n) (base 10), a(n) (base 2) - from N. J. A. Sloane, Sep 30 2018
# binary: binary representation of n, in human order
binary:=proc(n) local t1,L;
if n<0 then ERROR("n must be nonnegative"); fi;
if n=0 then return([0]); fi;
t1:=convert(n,base,2); L:=nops(t1);
[seq(t1[L+1-i],i=1..L)];
end;
for n from 0 to 100 do t1:=A003714(n); lprint(n, t1, binary(t1)); od:
  • Mathematica
    fibBin[n_Integer] := Block[{k = Ceiling[Log[GoldenRatio, n Sqrt[5]]], t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k], AppendTo[fr, 1]; t = t - Fibonacci[k], AppendTo[fr, 0]]; k--]; FromDigits[fr, 2]]; Table[fibBin[n], {n, 0, 61}] (* Robert G. Wilson v, Sep 18 2004 *)
Select[Range[0, 270], ! MemberQ[Partition[IntegerDigits[#, 2], 2, 1], {1, 1}] &] (* Harvey P. Dale, Jul 17 2011 *)
Select[Range[256], BitAnd[#, 2 #] == 0 &] (* Alonso del Arte, Jun 18 2012 *)
With[{r = Range[10^5]}, Pick[r, BitAnd[r, 2 r], 0]] (* Eric W. Weisstein, Aug 18 2017 *)
Select[Range[0, 299], SequenceCount[IntegerDigits[#, 2], {1, 1}] == 0 &] (* Requires Mathematica version 10 or later. -- Harvey P. Dale, Dec 06 2018 *)
  • PARI
    msb(n)=my(k=1); while(k<=n, k<<=1); k>>1
for(n=1,1e4,k=bitand(n,n<<1);if(k,n=bitor(n,msb(k)-1),print1(n", "))) \\ Charles R Greathouse IV, Jun 15 2011
  • PARI
    select( is_A003714(n)=!bitand(n,n>>1), [0..266])
{(next_A003714(n,t)=while(t=bitand(n+=1,n<<1), n=bitor(n,1<A003714(t)) \\ M. F. Hasler, Nov 30 2021
  • Python
    for n in range(300):
    if 2*n & n == 0:
        print(n, end=",") # Alex Ratushnyak, Jun 21 2012
  • Python
    def A003714(n):
    tlist, s = [1,2], 0
    while tlist[-1]+tlist[-2] <= n:
        tlist.append(tlist[-1]+tlist[-2])
    for d in tlist[::-1]:
        s *= 2
        if d <= n:
            s += 1
            n -= d
    return s # Chai Wah Wu, Jun 14 2018
  • Python
    def fibbinary():
    x = 0
    while True:
        yield x
        y = ~(x >> 1)
        x = (x - y) & y # Falk Hüffner, Oct 23 2021
(C++)
/* start with x=0, then repeatedly call x=next_fibrep(x): */
ulong next_fibrep(ulong x)
{
    // 2 examples:         //  ex. 1             //  ex.2
    //                     // x == [*]0 010101   // x == [*]0 01010
    ulong y = x | (x>>1);  // y == [*]? 011111   // y == [*]? 01111
    ulong z = y + 1;       // z == [*]? 100000   // z == [*]? 10000
    z = z & -z;            // z == [0]0 100000   // z == [0]0 10000
    x ^= z;                // x == [*]0 110101   // x == [*]0 11010
    x &= ~(z-1);           // x == [*]0 100000   // x == [*]0 10000
    return x;
}
/* Joerg Arndt, Jun 22 2012 */
  • Scala
    (0 to 255).filter(n => (n & 2 * n) == 0) // Alonso del Arte, Apr 12 2020
(C#)
public static bool IsFibbinaryNum(this int n) => ((n & (n >> 1)) == 0) ? true : false; // Frank Hollstein, Jul 07 2021

Formula

No two adjacent 1's in binary expansion.
Let f(x) := Sum_{n >= 0} x^Fibbinary(n). (This is the generating function of the characteristic function of this sequence.) Then f satisfies the functional equation f(x) = x*f(x^4) + f(x^2).
a(0) = 0, a(1) = 1, a(2) = 2, a(n) = 2^(A072649(n) - 1) + a(n - A000045(1 + A072649(n))). - Antti Karttunen
It appears that this sequence gives m such that A082759(3*m) is odd; or, probably equivalently, m such that A037011(3*m) = 1. - Benoit Cloitre, Jun 20 2003
If m is in the sequence then so are 2*m and 4*m + 1. - Henry Bottomley, Jan 11 2005
A116361(a(n)) <= 1. - Reinhard Zumkeller, Feb 04 2006
A085357(a(n)) = 1; A179821(a(n)) = a(n). - Reinhard Zumkeller, Jul 31 2010
a(n)/n^k is bounded (but does not tend to a limit), where k = 1.44... = A104287. - Charles R Greathouse IV, Sep 19 2012
a(n) = a(A193564(n+1))*2^(A003849(n) + 1) + A003849(n) for n > 0. - Daniel Starodubtsev, Aug 05 2021
There are Fibonacci(n+1) terms with up to n bits in this sequence. - Charles R Greathouse IV, Oct 22 2021
Sum_{n>=1} 1/a(n) = 3.704711752910469457886531055976801955909489488376627037756627135425780134020... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 12 2022

Extensions

Edited by Antti Karttunen, Feb 21 2006
Cross reference to A007820 added (into O-Y.C. comment) by Jason Kimberley, Sep 14 2009
Typo corrected by Jeffrey Shallit, Sep 26 2014

A292272 a(n) = n - A048735(n) = n - (n AND floor(n/2)).

Original entry on oeis.org

0, 1, 2, 2, 4, 5, 4, 4, 8, 9, 10, 10, 8, 9, 8, 8, 16, 17, 18, 18, 20, 21, 20, 20, 16, 17, 18, 18, 16, 17, 16, 16, 32, 33, 34, 34, 36, 37, 36, 36, 40, 41, 42, 42, 40, 41, 40, 40, 32, 33, 34, 34, 36, 37, 36, 36, 32, 33, 34, 34, 32, 33, 32, 32, 64, 65, 66, 66, 68, 69, 68, 68, 72, 73, 74, 74, 72, 73, 72, 72, 80, 81, 82, 82, 84, 85, 84, 84, 80, 81, 82, 82, 80, 81
Offset: 0

Views

Author

Antti Karttunen, Sep 16 2017

Keywords

Comments

In binary expansion of n, change those 1's to 0's that have an 1-bit next to them at their left (more significant) side. Only fibbinary numbers (A003714) occur as terms.

Examples

			From _Kevin Ryde_, Jun 02 2020: (Start)
     n = 1831 = binary 11100100111
  a(n) = 1060 = binary 10000100100   high 1 of each run
(End)

Links

Crossrefs

Cf. A003188, A003714, A003987, A004198, A005940, A048735, A085357, A292371, A292382.

Programs

Formula

a(n) = n - A048735(n) = n - (n AND floor(n/2)) = n XOR (n AND floor(n/2)), where AND is bitwise-AND (A004198) and XOR is bitwise-XOR (A003987).
a(n) = n AND A003188(n).
a(n) = A292382(A005940(1+n)).
A059905(a(n)) = A292371(n).
For all n >= 0, A085357(a(n)) = 1.
a(n) = A213064(n) / 2. - Kevin Ryde, Jun 02 2020
a(n) = n AND NOT floor(n/2). - Chai Wah Wu, Jun 29 2022

A132971 a(2*n) = a(n), a(4*n+1) = -a(n), a(4*n+3) = 0, with a(0) = 1.

Original entry on oeis.org

1, -1, -1, 0, -1, 1, 0, 0, -1, 1, 1, 0, 0, 0, 0, 0, -1, 1, 1, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 1, 0, 1, -1, 0, 0, 1, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 1, 0, 1, -1, 0, 0, 1, -1, -1, 0, 0, 0, 0, 0, 1, -1
Offset: 0

Views

Author

Michael Somos, Sep 17 2007, Sep 19 2007

Keywords

Comments

If binary(n) has adjacent 1 bits then a(n) = 0 else a(n) = (-1)^A000120(n).
Fibbinary numbers (A003714) gives the numbers n for which a(n) = A106400(n). - Antti Karttunen, May 30 2017

Examples

			G.f. = 1 - x - x^2 - x^4 + x^5 - x^8 + x^9 + x^10 - x^16 + x^17 + x^18 + ...

Links

Crossrefs

Cf. A000120, A000621, A003714, A005252, A005940, A008683, A024490, A027935, A106400.
Cf. A085357 (gives the absolute values: -1 -> 1), A286576 (when reduced modulo 3: -1 -> 2).

Programs

  • Mathematica
    m = 100; A[_] = 1;
Do[A[x_] = A[x^2] - x A[x^4] + O[x]^m // Normal, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Nov 16 2019 *)
  • PARI
    {a(n) = if( n<1, n==0, if( n%2, if( n%4 > 1, 0, -a((n-1)/4) ), a(n/2) ) )};
  • PARI
    {a(n) = my(A, m); if( n<0, 0, m = 1; A = 1 + O(x); while( m<=n, m *= 2; A = subst(A, x, x^2) - x * subst(A, x, x^4) ); polcoeff(A, n)) };
  • Python
    from sympy import mobius, prime, log
import math
def A(n): return n - 2**int(math.floor(log(n, 2)))
def b(n): return n + 1 if n<2 else prime(1 + (len(bin(n)[2:]) - bin(n)[2:].count("1"))) * b(A(n))
def a(n): return mobius(b(n)) # Indranil Ghosh, May 30 2017
  • Scheme
    (define (A132971 n) (cond ((zero? n) 1) ((even? n) (A132971 (/ n 2))) ((= 1 (modulo n 4)) (- (A132971 (/ (- n 1) 4)))) (else 0))) ;; Antti Karttunen, May 30 2017

Formula

A024490(n) = number of solutions to 2^n <= k < 2^(n+1) and a(k) = 1.
A005252(n) = number of solutions to 2^n <= k < 2^(n+1) and a(k) = -1.
A027935(n-1) = number of solutions to 2^n <= k < 2^(n+1) and a(k) = 0.
G.f. A(x) satisfies A(x) = A(x^2) - x * A(x^4).
G.f. B(x) of A000621 satisfies B(x) = x * A(x^2) / A(x).
a(n) = A008683(A005940(1+n)). [Analogous to Moebius mu] - Antti Karttunen, May 30 2017

A384715 a(n) = Sum_{k=0..n} (binomial(n, k) mod 4).

Original entry on oeis.org

1, 2, 4, 8, 4, 8, 12, 16, 4, 8, 12, 24, 12, 24, 24, 32, 4, 8, 12, 24, 12, 24, 32, 48, 12, 24, 32, 48, 24, 48, 48, 64, 4, 8, 12, 24, 12, 24, 32, 48, 12, 24, 32, 64, 32, 64, 64, 96, 12, 24, 32, 48, 32, 64, 64, 96, 24, 48, 64, 96, 48, 96, 96, 128, 4, 8, 12, 24
Offset: 0

Views

Author

David Radcliffe, Jun 23 2025

Keywords

Comments

This is a 2-automatic sequence.

Examples

			Let b(n) be the binary expansion of n. Then a(n) = (1 + p10 + p11) * 2^c, where c is the number of set bits in b(n), p10 is the number of '10' patterns in b(n), and p11 is 1 or 0 depending on whether the pattern '11' is occurring in b(n) or not. This formula is used by _Chai Wah Wu_ in the Python function below. For instance:
  n = 25 -> b(n) = 11001 -> a(n) = (1+1+1) * 2^3 = 24.
  n = 26 -> b(n) = 11010 -> a(n) = (1+2+1) * 2^3 = 32.
  n = 27 -> b(n) = 11011 -> a(n) = (1+1+1) * 2^4 = 48.
- _Peter Luschny_, Jun 25 2025

Links

Crossrefs

Cf. A001316, A014081, A033264, A051638 (mod 3 analog), A085357. Row sums of triangle A034931.

Programs

  • Mathematica
    A384715[n_] := 2^DigitSum[n, 2]*(StringCount[IntegerString[n, 2], "10"] - Boole[BitAnd[n,2*n] == 0] + 2);
Array[A384715, 100, 0] (* Paolo Xausa, Jun 26 2025 *)
  • PARI
    a(n) = sum(k=0, n, binomial(n,k)%4); \\ Michel Marcus, Jun 25 2025
  • Python
    def A001316(n): return (1 + (n % 2)) * A001316(n // 2) if n else 1
def A033264(n): return (n % 4 == 2) + A033264(n // 2) if n else 0
def A085357(n): return int(n & (n<<1) == 0)
def A384715(n): return A001316(n) * (A033264(n) - A085357(n) + 2)
  • Python
    def A384715(n): return (((n>>1)&~n).bit_count()+bool(n&(n<<1))+1)<Chai Wah Wu, Jun 25 2025
  • Python
    def a(n: int) -> int:  # after Chai Wah Wu
    b = bin(n)[2:]; p = b.count("10"); q = b.count("11")
    return (p + (2 if q else 1)) * 2**n.bit_count()  # Peter Luschny, Jun 25 2025

Formula

a(n) = A001316(n) * (A033264(n) - A085357(n) + 2) for n > 0.
Recurrences:
a(4n) = a(2n),
a(4n+1) = 2a(2n),
a(8n+2) = a(4n+2) + 2a(2n) - a(2n+1),
a(8n+3) = a(4n+3) + 4a(2n) - 4a(n),
a(8n+6) = a(4n+3) + 2a(4n+2) - 2a(2n+1),
a(8n+7) = 2a(4n+3).

A085405 Common residues of binomial(3n+2,n+1)/(3n+2) modulo 2.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Paul D. Hanna, Jun 29 2003

Keywords

Comments

The positions of ones are given by A022340 and runs of zeros are given by A085407: both are related to the Fibonacci sequence.

Links

Crossrefs

Cf. A005940, A006013, A022340 (ones), A085407 (zeros), A085357, A277332, A323239.

Programs

  • PARI
    A085405(n) = ((binomial((3*n)+2, n+1)/((3*n)+2))%2); \\ Antti Karttunen, Jan 12 2019
  • PARI
    A085405(n) = if(n%2,0,while(n>0, my(nextn=(n>>1)); if(1==(nextn%2)*(n%2), return(0)); n = nextn); (1)); \\ (Much faster than above program) - Antti Karttunen, Jan 12 2019

Formula

a(n) = C(3n+2, n+1)/(3n+2) (Mod 2) = A006013(n) (Mod 2), where A006013 is the self-convolution of A001764 (ternary trees).
a(n) = A323239(A005940(1+n)). - Antti Karttunen, Jan 12 2019

A090077 In binary expansion of n: reduce contiguous blocks of 1's to 1.

Original entry on oeis.org

0, 1, 2, 1, 4, 5, 2, 1, 8, 9, 10, 5, 4, 5, 2, 1, 16, 17, 18, 9, 20, 21, 10, 5, 8, 9, 10, 5, 4, 5, 2, 1, 32, 33, 34, 17, 36, 37, 18, 9, 40, 41, 42, 21, 20, 21, 10, 5, 16, 17, 18, 9, 20, 21, 10, 5, 8, 9, 10, 5, 4, 5, 2, 1, 64, 65, 66, 33, 68, 69, 34, 17, 72, 73, 74, 37, 36, 37, 18, 9
Offset: 0

Views

Author

Reinhard Zumkeller, Nov 20 2003

Keywords

Examples

			100 -> '1100100' -> [11]00[1]00 -> [1]00[1]00 -> '100100' -> 36=a(100).

Links

Crossrefs

Cf. A007088, A090079, A090078, A348710.

Programs

  • Mathematica
    Array[FromDigits[Flatten[Split@ IntegerDigits[#, 2] /. w_List /; First[w] == 1 -> {1}], 2] &, 80, 0] (* Michael De Vlieger, Jul 28 2022 *)
  • Python
    def a(n):
    b = bin(n)[2:]
    while "11" in b: b = b.replace("11", "1")
    return int(b, 2)
print([a(n) for n in range(81)]) # Michael S. Branicky, Jul 27 2022

Formula

a(a(n)) = a(n); a(A090078(n)) = A090078(a(n)) = A090079(n).
a(A003714(n)) = A003714(n); a(A004780(n)) < A004780(n); a(n) <= A179821(n); A085357(a(n)) = 1. - Reinhard Zumkeller, Jul 31 2010

A153013 Starting with input 0, find the binary value of the input. Then interpret resulting string of 1's and 0's as prime-based numbers, as follows: 0's are separators, uninterrupted strings of 1's are interpreted from right to left as exponents of the prime numbers. Output is returned as input for the next number in sequence.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 9, 10, 15, 16, 11, 12, 25, 50, 147, 220, 6125, 1968750, 89142864525, 84252896510182189218, 34892570216750728458698250328871491829901861750593684043
Offset: 0

Views

Author

Mark Zegarelli (mtzmtz(AT)gmail.com), Dec 16 2008

Keywords

Comments

From Antti Karttunen, Oct 15 2016: (Start)
Iterates of map f : n -> A005940(1+n), (Doudna-sequence, but with starting offset zero) starting from the initial value 0. Conversely, the unique infinite sequence such that a(n) = A156552(a(n+1)) and a(0) = 0.
Note that map f can also form cycles, like 7 <-> 8 (A005940(1+7) = 8, A005940(1+8) = 7).
On the other hand, this sequence cannot ever fall into a loop because 0 is not in the range of map f, for n=0.., while f is injective on [1..]. Thus the values obtained by this sequence are not bounded, although there might be more nonmonotonic positions like for example there is from a(10) = 16 to a(11) = 11.
The formula A008966(a(n+1)) = A085357(a(n)) tells that the squarefreeness of the next term a(n+1) is determined by whether the previous term a(n) is a fibbinary number (A003714) or not. Numerous other such correspondences hold, and they hold also for any other trajectories outside of this sequence.
Even and odd terms alternate. No two squares can occur in succession because A106737 obtains even values for all squares > 1 and A000005 is odd for all squares. More directly this is seen from the fact that the rightmost 1-bit in the binary expansion of any square is always alone.
(End)

Examples

			101 is interpreted as 3^1 * 2^1 = 6. 1110011 is interpreted as 5^3 * 2^2 = 500.

Links

Crossrefs

Cf. A000005, A003714, A005940, A008966, A085357, A106737, A156552, A181819, A246029.
Cf. also A109162, A328316 for similar iteration sequences.

Programs

  • Mathematica
    NestList[Times @@ Flatten@ MapIndexed[Prime[#2]^#1 &, #] &@ Flatten@ MapIndexed[If[Total@ #1 == 0, ConstantArray[0, Boole[First@ #2 == 1] + Length@ #1 - 1], Length@ #1] &, Reverse@ Split@ IntegerDigits[#, 2]] &, 0, 21] (* Michael De Vlieger, Oct 17 2016 *)
  • PARI
    step(n)=my(t=1,v); forprime(p=2,, v=valuation(n+1,2); t*=p^v; n>>=v+1; if(!n, return(t)))
t=0; concat(0,vector(20,n, t=step(t))) \\ Charles R Greathouse IV, Sep 01 2015
  • Scheme
    ;; With memoization-macro definec.
(definec (A153013 n) (if (zero? n) n (A005940 (+ 1 (A153013 (- n 1))))))
;; Antti Karttunen, Oct 15 2016

Formula

From Antti Karttunen, Oct 15 2016: (Start)
a(0) = 0; for n >= 1, a(n) = A005940(1+a(n-1)).
A008966(a(n+1)) = A085357(a(n)). [See the comment.]
A181819(a(1+n)) = A246029(a(n)).
A000005(a(n+1)) = A106737(a(n)).
(End)

Extensions

a(20)-a(22) from Yang Haoran, Aug 31 2015
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