A003188 -id:A003188 - cp's OEIS frontend

A286555 Permutation of natural numbers: a(n) = A252753(A003188(n)).

1, 2, 4, 3, 9, 8, 6, 5, 25, 18, 16, 21, 15, 12, 10, 7, 49, 50, 36, 51, 45, 32, 42, 55, 35, 30, 24, 33, 27, 20, 14, 11, 121, 98, 100, 147, 105, 72, 102, 125, 115, 90, 64, 93, 123, 84, 110, 91, 77, 70, 60, 87, 69, 48, 66, 85, 65, 54, 40, 57, 39, 28, 22, 13, 169, 242, 196, 291, 297, 200, 294, 365, 265, 210, 144, 213

Offset: 0

Antti Karttunen, May 13 2017

nonn

Note the indexing: the domain starts from 0, but the range from 1.

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences that are permutations of the natural numbers

Inverse: A286556.
Cf. A286557.
Related or similar permutations: A003188, A252753.

(define (A286555 n) (A252753 (A003188 n)))

a(n) = A252753(A003188(n)).

A046523(a(n)) = A286557(n).

A302784 Inverse permutation to A302783: a(n) = A003188(A052331(n)).

Original entry on oeis.org

0, 1, 3, 6, 12, 2, 24, 7, 48, 13, 96, 5, 192, 25, 15, 384, 768, 49, 1536, 10, 27, 97, 3072, 4, 6144, 193, 51, 30, 12288, 14, 24576, 385, 99, 769, 20, 54, 49152, 1537, 195, 11, 98304, 26, 196608, 102, 60, 3073, 393216, 387, 786432, 6145, 771, 198, 1572864, 50, 108, 31, 1539, 12289, 3145728, 9, 6291456, 24577, 40, 390, 204, 98, 12582912, 774, 3075, 21

Offset: 1

Antti Karttunen, Apr 16 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..4096
Index entries for sequences that are permutations of the natural numbers

Cf. A302783 (inverse).

Cf. also A302029.

up_to = 4096;
v050376 = vector(up_to);
ispow2(n) = (n && !bitand(n,n-1));
i = 0; for(n=1,oo,if(ispow2(isprimepower(n)), i++; v050376[i] = n); if(i == up_to,break));
A052331(n) = { my(s=0,e); while(n > 1, fordiv(n, d, if(((n/d)>1)&&ispow2(isprimepower(n/d)), e = vecsearch(v050376, n/d); if(!e, print("v050376 too short!"); return(1/0)); s += 2^(e-1); n = d; break))); (s); };
A003188(n) = bitxor(n, n>>1);
A302784(n) = A003188(A052331(n));

a(n) = A003188(A052331(n)).

A322015 If A003188(n+1) < A003188(n), then a(n) = n+1, otherwise a(n) = 0.

Original entry on oeis.org

0, 0, 3, 0, 0, 6, 7, 0, 0, 0, 11, 12, 0, 14, 15, 0, 0, 0, 19, 0, 0, 22, 23, 24, 0, 0, 27, 28, 0, 30, 31, 0, 0, 0, 35, 0, 0, 38, 39, 0, 0, 0, 43, 44, 0, 46, 47, 48, 0, 0, 51, 0, 0, 54, 55, 56, 0, 0, 59, 60, 0, 62, 63, 0, 0, 0, 67, 0, 0, 70, 71, 0, 0, 0, 75, 76, 0, 78, 79, 0, 0, 0, 83, 0, 0, 86, 87, 88, 0, 0, 91, 92, 0, 94, 95, 96

Offset: 0

Antti Karttunen, Nov 24 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for sequences related to binary expansion of n

Cf. A003188, A322016, A322017.

Cf. also A091067, A255068.

With[{nn = 96}, ReplacePart[ConstantArray[0, nn], Map[# -> # &, 1 + Position[Partition[Array[BitXor[#, Floor[#/2]] &, nn], 2, 1], ?(First@ # > Last@ # &), {1}, Heads -> False][[All, 1]] ]]] (* _Michael De Vlieger, Nov 25 2018 *)

A003188(n) = bitxor(n, n>>1);
A322015(n) = if(A003188(1+n)<A003188(n),1+n,0);

A322016 a(0) = 0; for n > 0, if A003188(n) > A003188(n-1) then a(n) = n-1, otherwise a(n) = 0.

Original entry on oeis.org

0, 0, 1, 0, 3, 4, 0, 0, 7, 8, 9, 0, 0, 12, 0, 0, 15, 16, 17, 0, 19, 20, 0, 0, 0, 24, 25, 0, 0, 28, 0, 0, 31, 32, 33, 0, 35, 36, 0, 0, 39, 40, 41, 0, 0, 44, 0, 0, 0, 48, 49, 0, 51, 52, 0, 0, 0, 56, 57, 0, 0, 60, 0, 0, 63, 64, 65, 0, 67, 68, 0, 0, 71, 72, 73, 0, 0, 76, 0, 0, 79, 80, 81, 0, 83, 84, 0, 0, 0, 88, 89, 0, 0, 92, 0, 0, 0, 96, 97

Offset: 0

Antti Karttunen, Nov 24 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for sequences related to binary expansion of n

Cf. A003188, A034947, A322015, A322017.

With[{nn = 98}, {0, 0}~Join~Most@ ReplacePart[ConstantArray[0, nn], Map[# -> # &, Position[Partition[Array[BitXor[#, Floor[#/2]] &, nn], 2, 1], ?(First@ # < Last@ # &), {1}, Heads -> False][[All, 1]] ]]] (* _Michael De Vlieger, Nov 25 2018 *)
Join[{0,0},With[{t=Partition[Table[BitXor[n,Floor[n/2]],{n,100}],2,1]},Flatten[ If[#[[1]]<#[[2]],Position[t,#],0]&/@t]]] (* Harvey P. Dale, Jan 28 2021 *)

A003188(n) = bitxor(n, n>>1);
A322016(n) = if(0==n,0,if(A003188(n)>A003188(n-1),n-1,0));

A322016(n) = if(!n,0,(((kronecker(-1,n)+1)/2)*(n-1)));

a(0) = 0; for n > 0, a(n) = (1/2)*(A034947(n)+1)*(n-1).

A099892 XOR BINOMIAL transform of A003188 (Gray code numbers); also the main diagonal of the XOR difference triangle A099891.

Original entry on oeis.org

0, 1, 3, 0, 6, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 48, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 96, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Paul D. Hanna, Oct 29 2004

See A099884 for the definitions of the XOR BINOMIAL transform and the XOR difference triangle.

Cf. A099884, A003188, A099891.

a[n_] := Module[{e = IntegerExponent[n, 2]}, Switch[n, 0, 0, 1, 1, 2^e, 3*2^(e - 1), , 0]]; Array[a, 100, 0] (* _Amiram Eldar, Aug 31 2023, corrected by Michael Shamos, May 22 2025 *)

{a(n)=local(B);B=0;for(i=0,n,B=bitxor(B,binomial(n,i)%2*(bitxor((n-i),(n-i)\2))));B}

a(2^n) = 3*2^(n-1) for n>0, with a(0)=0, a(1) = 1 and a(k)=0 otherwise. a(n) = SumXOR_{i=0..n} (C(n, i)mod 2)*A003188(n-i), where A003188(k)=bitxor(k, [k/2]) and SumXOR is summation under XOR.
Multiplicative with a(2^e) = 3*2^(e-1), a(p^e) = 0 otherwise. - David W. Wilson, Jun 12 2005
Dirichlet g.f.: (2^s+1)/(2^s-2). - R. J. Mathar, Apr 14 2011

A227526 G.f.: Sum_{n>=0} x^n * (1+x)^A003188(n), where A003188(n) = n XOR [n/2] is the Gray code for n.

Original entry on oeis.org

1, 1, 2, 4, 6, 9, 23, 47, 65, 70, 111, 323, 902, 2064, 3997, 6697, 9638, 11664, 11804, 11436, 18903, 61053, 209236, 623648, 1610326, 3685365, 7673031, 14926689, 27784642, 50202979, 88022528, 147757232, 232939294, 338735927, 448199945, 534600063, 571408283, 546281026, 475461610

Offset: 0

Paul D. Hanna, Jul 14 2013

nonn

A(-1/2) = Sum_{n>=0} (-1)^n / 2^(n + A003188(n)) = 0.75073331594540770872470210...

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 6*x^4 + 9*x^5 + 23*x^6 + 47*x^7 + 65*x^8 +...
where
A(x) = 1 + x*(1+x) + x^2*(1+x)^3 + x^3*(1+x)^2 + x^4*(1+x)^6 + x^5*(1+x)^7 + x^6*(1+x)^5 + x^7*(1+x)^4 + x^8*(1+x)^12 + x^9*(1+x)^13 + x^10*(1+x)^15 + x^11*(1+x)^14 + x^12*(1+x)^10 + x^13*(1+x)^11 + x^14*(1+x)^9 + x^15*(1+x)^8 +...

Paul D. Hanna, Table of n, a(n) for n = 0..1024

Cf. A227527, A003188.

{a(n)=polcoeff(sum(m=0, n, x^m*(1+x+x*O(x^n))^bitxor(m,m\2)), n)}
for(n=0, 64, print1(a(n), ", "))

{a(n) = sum(k=0, n, binomial(bitxor(k,k\2), n-k))}
for(n=0, 64, print1(a(n), ", "))

a(n) = Sum_{k=0..n} binomial(A003188(k), n-k), where A003188(k) = k XOR [k/2].

A227527 G.f.: Sum_{n>=0} x^n * (1-x)^A003188(n), where A003188(n) = n XOR [n/2] is the Gray code for n.

Original entry on oeis.org

1, 1, 0, -2, 2, -5, 9, -3, -13, 14, 35, -149, 300, -450, 673, -1151, 1856, -2366, 2424, -3192, 9319, -32687, 96858, -238410, 508290, -998065, 1925065, -3750685, 7162328, -12635545, 19437562, -24482990, 22154946, -6283107, -25823457, 69598597, -113006459, 140737910, -137007218, 67953174

Offset: 0

Paul D. Hanna, Jul 14 2013

sign

A(1/2) = Sum_{n>=0} 1/2^(n + A003188(n)) = 1.3146990537656455533266364007421...

			G.f.: A(x) = 1 + x - 2*x^3 + 2*x^4 - 5*x^5 + 9*x^6 - 3*x^7 - 13*x^8 + 14*x^9 +...
where A(x) = A(1-x) equals the series:
A(x) = 1 + x*(1-x) + x^2*(1-x)^3 + x^3*(1-x)^2 + x^4*(1-x)^6 + x^5*(1-x)^7 + x^6*(1-x)^5 + x^7*(1-x)^4 + x^8*(1-x)^12 + x^9*(1-x)^13 + x^10*(1-x)^15 + x^11*(1-x)^14 + x^12*(1-x)^10 + x^13*(1-x)^11 + x^14*(1-x)^9 + x^15*(1-x)^8 +...

Paul D. Hanna, Table of n, a(n) for n = 0..1024

Cf. A227526, A003188.

{a(n)=polcoeff(sum(m=0, n, x^m*(1-x+x*O(x^n))^bitxor(m,m\2)), n)}
for(n=0, 64, print1(a(n), ", "))

{a(n) = sum(k=0, n, (-1)^(n-k)*binomial(bitxor(k,k\2), n-k))}
for(n=0, 64, print1(a(n), ", "))

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(A003188(k), n-k), where A003188(k) = k XOR [k/2].

A268722 a(n) = A003188(3*A006068(n)), where A003188 is binary Gray code and A006068 is its inverse.

Original entry on oeis.org

0, 2, 13, 5, 31, 27, 10, 8, 59, 63, 54, 52, 20, 22, 49, 17, 115, 119, 126, 124, 108, 110, 121, 105, 40, 42, 37, 45, 103, 99, 34, 32, 227, 231, 238, 236, 252, 254, 233, 249, 216, 218, 213, 221, 247, 243, 210, 208, 80, 82, 93, 85, 79, 75, 90, 88, 203, 207, 198, 196, 68, 70, 193, 65

Offset: 0

Antti Karttunen, Feb 13 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8191

Cf. A003188, A006068.

Row 2 and column 2 of array A268725.

def A268722(n):
    k, m = n, n>>1
    while m > 0:
        k ^= m
        m >>= 1
    return (a:=3*k)^ a>>1 # Chai Wah Wu, Jun 30 2022

(define (A268722 n) (A003188 (* 3 (A006068 n))))

a(n) = A003188(3*A006068(n)).

A268723 Main diagonal of A268725: a(n) = A003188(A006068(n)^2), where A003188 is binary Gray code and A006068 is its inverse.

Original entry on oeis.org

0, 1, 13, 6, 41, 54, 24, 21, 145, 166, 216, 253, 96, 121, 69, 86, 545, 582, 664, 749, 864, 841, 949, 1014, 384, 433, 477, 486, 793, 278, 344, 357, 2113, 2182, 2328, 2509, 2656, 2793, 2901, 2998, 3456, 3537, 3901, 3366, 3641, 3798, 4056, 3973, 1536, 1633, 1709, 1734, 1801, 1910, 1944, 2037, 3313, 3174, 1112, 1053

Offset: 0

Antti Karttunen, Feb 13 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8191

Cf. A000290, A003188, A006068.

Main diagonal of array A268725.

def A268723(n):
    k, m = n, n>>1
    while m > 0:
        k ^= m
        m >>= 1
    return (a:=k**2)^ a>>1 # Chai Wah Wu, Jun 30 2022

(define (A268723 n) (A003188 (A000290 (A006068 n))))

a(n) = A003188(A000290(A006068(n))).

A286383 a(n) = A278233(A003188(n)).

Original entry on oeis.org

1, 2, 2, 6, 2, 4, 4, 12, 2, 8, 6, 12, 2, 6, 8, 24, 2, 12, 6, 24, 2, 6, 12, 36, 4, 6, 6, 30, 2, 16, 16, 48, 6, 32, 6, 60, 2, 6, 12, 72, 2, 12, 6, 30, 2, 12, 24, 72, 2, 6, 12, 30, 2, 24, 12, 60, 2, 12, 6, 48, 6, 6, 32, 96, 2, 12, 30, 96, 2, 30, 12, 180, 2, 6, 6, 30, 8, 24, 24, 216, 6, 6, 6, 60, 6, 12, 12, 60, 2, 48, 6, 60, 2, 6, 48, 144, 4, 30, 6, 30, 2, 64, 36

Offset: 1

Antti Karttunen, May 08 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16383
Index entries for sequences related to binary expansion of n

Cf. A003188, A278233, A286384.

(define (A286383 n) (A278233 (A003188 n)))

a(n) = A278233(A003188(n)).

cp's OEIS Frontend

A286555 Permutation of natural numbers: a(n) = A252753(A003188(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Scheme

Formula

A302784 Inverse permutation to A302783: a(n) = A003188(A052331(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A322015 If A003188(n+1) < A003188(n), then a(n) = n+1, otherwise a(n) = 0.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A322016 a(0) = 0; for n > 0, if A003188(n) > A003188(n-1) then a(n) = n-1, otherwise a(n) = 0.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A099892 XOR BINOMIAL transform of A003188 (Gray code numbers); also the main diagonal of the XOR difference triangle A099891.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Formula

A227526 G.f.: Sum_{n>=0} x^n * (1+x)^A003188(n), where A003188(n) = n XOR [n/2] is the Gray code for n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

A227527 G.f.: Sum_{n>=0} x^n * (1-x)^A003188(n), where A003188(n) = n XOR [n/2] is the Gray code for n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

A268722 a(n) = A003188(3*A006068(n)), where A003188 is binary Gray code and A006068 is its inverse.

Views

Author

Keywords

Links

Crossrefs