Nick Krempel - cp's OEIS frontend

A332868 Number of involutions (plus identity) in a fixed Sylow 2-subgroup of the symmetric group of degree 2n.

1, 2, 6, 12, 44, 88, 264, 528, 2064, 4128, 12384, 24768, 90816, 181632, 544896, 1089792, 4292864, 8585728, 25757184, 51514368, 188886016, 377772032, 1133316096, 2266632192, 8860471296, 17720942592, 53162827776, 106325655552, 389860737024, 779721474048, 2339164422144

Offset: 0

Nick Krempel, Feb 27 2020

nonn

Bisection of A332759.

			For n=2, the a(2)=6 elements satisfying x^2=1 in a fixed Sylow 2-subgroup of S_4 (which subgroup is isomorphic to the dihedral group of degree 4) are the identity and (13), (24), (12)(34), (13)(24), (14)(23).

Alois P. Heinz, Table of n, a(n) for n = 0..2412

Cf. A332757, A332759.

b:= proc(n) b(n):=`if`(n=0, 1, b(n-1)^2+2^(2^(n-1)-1)) end:
a:= n-> (l-> mul(`if`(l[i]=1, b(i), 1), i=1..nops(l)))(Bits[Split](n)):
seq(a(n), n=0..35);  # Alois P. Heinz, Feb 27 2020

b[n_] := b[n] = If[n == 0, 1, b[n - 1]^2 + 2^(2^(n - 1) - 1)];
a[n_] := Function[l, Product[If[l[[i]] == 1, b[i], 1], {i, 1, Length[l]}]][ Reverse @ IntegerDigits[n, 2]];
a /@ Range[0, 35] (* Jean-François Alcover, Apr 10 2020, after Alois P. Heinz *)

a(n)={my(v=vector(logint(max(1,n), 2)+1)); v[1]=2; for(n=2, #v, v[n]=v[n-1]^2 + 2^(2^(n-1)-1)); prod(k=1, #v, if(bittest(n,k-1), v[k], 1))} \\ Andrew Howroyd, Feb 27 2020

a(n) = A332759(2*n).
a(n) = Product(A332757(k+1)) where k ranges over the positions of 1 bits in the binary expansion of n.
a(n) = big-Theta(C^n) for C = 2.59745646488..., i.e., A*C^n < a(n) < B*C^n for constants A, B (but it's not the case that a(n) ~ C^n as lim inf a(n)/C^n and lim sup a(n)/C^n differ).

Terms a(17) and beyond from Andrew Howroyd, Feb 27 2020

A332869 Number of fixed-point free involutions in a fixed Sylow 2-subgroup of the symmetric group of degree 4n.

Original entry on oeis.org

1, 3, 17, 51, 417, 1251, 7089, 21267, 206657, 619971, 3513169, 10539507, 86175969, 258527907, 1464991473, 4394974419, 44854599297, 134563797891, 762528188049, 2287584564147, 18704367906849, 56113103720547, 317974254416433, 953922763249299, 9269516926920129

Offset: 0

Nick Krempel, Feb 27 2020

nonn

Bisection of A332840.

			For n=1, the a(1)=3 fixed-point free involutions in a fixed Sylow 2-subgroup of S_4 (which subgroup is isomorphic to the dihedral group of degree 4) are (12)(34), (13)(24), and (14)(23).

Alois P. Heinz, Table of n, a(n) for n = 0..1502

Cf. A332758, A332840.

b:= proc(n) b(n):=`if`(n=0, 0, b(n-1)^2+2^(2^(n-1)-1)) end:
a:= n-> (l-> mul(`if`(l[i]=1, b(i+1), 1), i=1..nops(l)))(Bits[Split](n)):
seq(a(n), n=0..32);  # Alois P. Heinz, Feb 27 2020

A332758[n_] := A332758[n] = If[n==0, 0, A332758[n-1]^2 + 2^(2^(n-1)-1)];
a[n_] := Product[A332758[k+1], {k, Flatten@ Position[ Reverse@ IntegerDigits[n, 2], 1]}];
a /@ Range[0, 24] (* Jean-François Alcover, Apr 10 2020 *)

a(n)={my(v=vector(logint(max(1,n), 2)+2)); v[1]=1; for(n=2, #v, v[n]=v[n-1]^2 + 2^(2^(n-1)-1)); prod(k=2, #v, if(bittest(n,k-2), v[k], 1))} \\ Andrew Howroyd, Feb 27 2020

a(n) = A332840(2*n).
a(n) = Product(A332758(k+2)) where k ranges over the positions of 1 bits in the binary expansion of n.
a(n) = big-Theta(C^n) for C = 4.63233857..., i.e., A*C^n < a(n) < B*C^n for constants A, B (but it's not the case that a(n) ~ C^n as lim inf a(n)/C^n and lim sup a(n)/C^n differ).

Terms a(9) and beyond from Andrew Howroyd, Feb 27 2020

A332840 Number of fixed-point free involutions in a fixed Sylow 2-subgroup of the symmetric group of degree 2n.

Original entry on oeis.org

1, 1, 3, 3, 17, 17, 51, 51, 417, 417, 1251, 1251, 7089, 7089, 21267, 21267, 206657, 206657, 619971, 619971, 3513169, 3513169, 10539507, 10539507, 86175969, 86175969, 258527907, 258527907, 1464991473, 1464991473, 4394974419, 4394974419, 44854599297, 44854599297, 134563797891

Offset: 0

Nick Krempel, Feb 26 2020

nonn

As a Sylow 2-subgroup of S_(4n+2) is isomorphic to a Sylow 2-subgroup of S_(4n) direct product C_2, the terms of this sequence come in equal pairs.

Also the number of fixed-point free involutory automorphisms of the full binary tree with 2n leaves (hence 4n-1 vertices) in which all left children are complete (perfect) binary trees.

			For n=2, the a(2)=3 fixed-point free involutions in a fixed Sylow 2-subgroup of S_4 (which subgroup is isomorphic to the dihedral group of degree 4) are (12)(34), (13)(24), and (14)(23).

Cf. A123023, A001147, A332758, A332869.

b:= proc(n) b(n):=`if`(n=0, 0, b(n-1)^2+2^(2^(n-1)-1)) end:
a:= n-> (l-> mul(`if`(l[i]=1, b(i), 1), i=1..nops(l)))(Bits[Split](n)):
seq(a(n), n=0..40);  # Alois P. Heinz, Feb 27 2020

A332758[n_] := A332758[n] = If[n == 0, 0, A332758[n-1]^2 + 2^(2^(n-1)-1)];
a[n_] := Product[A332758[k], {k, Flatten@ Position[ Reverse@ IntegerDigits[ n, 2], 1]}];
a /@ Range[0, 34] (* Jean-François Alcover, Apr 10 2020 *)

a(n)={my(v=vector(logint(max(1,n), 2)+1)); v[1]=1; for(n=2, #v, v[n]=v[n-1]^2 + 2^(2^(n-1)-1)); prod(k=1, #v, if(bittest(n,k-1), v[k], 1))} \\ Andrew Howroyd, Feb 27 2020

a(n) = Product(A332758(k+1)) where k ranges over the positions of 1 bits in the binary expansion of n.
a(n) = big-Theta(C^n) for C = 2.1522868238..., i.e., A*C^n < a(n) < B*C^n for constants A, B (but it's not the case that a(n) ~ C^n as lim inf a(n)/C^n and lim sup a(n)/C^n differ).
a(n) = A332869(floor(n/2)). - Andrew Howroyd, Feb 27 2020

Terms a(18) and beyond from Andrew Howroyd, Feb 27 2020

A332759 Number of involutions (plus identity) in a fixed Sylow 2-subgroup of the symmetric group of degree n.

Original entry on oeis.org

1, 1, 2, 2, 6, 6, 12, 12, 44, 44, 88, 88, 264, 264, 528, 528, 2064, 2064, 4128, 4128, 12384, 12384, 24768, 24768, 90816, 90816, 181632, 181632, 544896, 544896, 1089792, 1089792, 4292864, 4292864, 8585728, 8585728, 25757184, 25757184, 51514368, 51514368

Offset: 0

Nick Krempel, Feb 22 2020

nonn

As the Sylow 2-subgroups of S_(2n) are isomorphic to those of S_(2n+1), the terms of this sequence come in pairs.

Also the number of involutory automorphisms (including identity) of the full binary tree with n leaves (hence 2n-1 vertices) in which all left children are complete (perfect) binary trees.

			For n=4, the a(4)=6 elements satisfying x^2=1 in a fixed Sylow 2-subgroup of S_4 (which subgroup is isomorphic to the dihedral group of degree 4) are the identity and (13), (24), (12)(34), (13)(24), (14)(23).

Cf. A000085.

b:= proc(n) b(n):=`if`(n=0, 1, b(n-1)^2+2^(2^(n-1)-1)) end:
a:= n-> (l-> mul(`if`(l[i]=1, b(i-1), 1), i=1..nops(l)))(Bits[Split](n)):
seq(a(n), n=0..50);  # Alois P. Heinz, Feb 27 2020

Join[{1}, Block[{nn = 33, s}, s = Nest[Append[#1, #1[[-1]]^2 + 2^(2^(#2 - 1) - 1)] & @@ {#, Length@ #} &, {1}, Ceiling@ Log2@ nn]; Array[Times @@ s[[Position[Reverse@ IntegerDigits[#, 2], 1][[All, 1]] ]] &, nn]]] (* Michael De Vlieger, Feb 25 2020 *)

a(n) = Product(A332757(k)) where k ranges over the positions of 1 bits in the binary expansion of n.
a(n) = big-Theta(C^n) for C = 1.6116626399..., i.e., A*C^n < a(n) < B*C^n for constants A, B (but it's not the case that a(n) ~ C^n as lim inf a(n)/C^n and lim sup a(n)/C^n differ).
Conjecture: B=1 and A=0.409091077245262341747187571213565366725933766222357989... - Vaclav Kotesovec, Feb 26 2020

More terms from Alois P. Heinz, Feb 27 2020

A332757 Number of involutions (plus identity) in the n-fold iterated wreath product of C_2.

Original entry on oeis.org

1, 2, 6, 44, 2064, 4292864, 18430828806144, 339695459704759501186924544, 115393005344028056118476170527365821430429589033713664, 13315545682326887517994506072805639054664915214679444711916992466809542959290217586307654871548759705124864

Offset: 0

Nick Krempel, Feb 22 2020

nonn

Also the number of involutions (plus identity) in a fixed Sylow 2-subgroup of the symmetric group of degree 2^n.

Also the number of involutory automorphisms (including identity) of the complete binary tree of height n.

			For n=2, the a(2)=6 elements satisfying x^2=1 in C_2 wr C_2 (which is isomorphic to the dihedral group of degree 4) are the identity and (13), (24), (12)(34), (13)(24), (14)(23).

Claudia Scheimbauer and Thomas Stempfhuber, Relative field theories via relative dualizability, arXiv:2312.05051 [math.CT], 2023. See Theorem 6.7 at pages 5, 32, and 34.

Cf. A332758.

Nest[Append[#1, #1[[-1]]^2 + 2^(2^(#2 - 1) - 1)] & @@ {#, Length@ #} &, {1}, 9] (* Michael De Vlieger, Feb 25 2020 *)

a(n) = a(n-1)^2 + 2^(2^(n-1)-1), a(0) = 1.

a(n) ~ C^(2^n) for C = 1.611662639909645505576094683462403213269601342091954838587...

A332758 Number of fixed-point free involutions in the n-fold iterated wreath product of C_2.

Original entry on oeis.org

0, 1, 3, 17, 417, 206657, 44854599297, 2021158450131287670017, 4085251621720569336520310526902208564886017, 16689280870666586360302304039420036318743515355074220606298783584912362351240766944257

Offset: 0

Nick Krempel, Feb 22 2020

nonn

Also the number of fixed-point free involutions in a fixed Sylow 2-subgroup of the symmetric group of degree 2^n.

Also the number of fixed-point free involutory automorphisms of the complete binary tree of height n.

			For n=2, the a(2)=3 fixed-point free involutions in C_2 wr C_2 (which is isomorphic to the dihedral group of degree 4) are (12)(34), (13)(24), and (14)(23).

Cf. A332757.

Nest[Append[#1, #1[[-1]]^2 + 2^(2^(#2 - 1) - 1)] & @@ {#, Length@ #} &, {0}, 9] (* Michael De Vlieger, Feb 25 2020 *)

a(n) = a(n-1)^2 + 2^(2^(n-1)-1), a(0) = 0.
a(n) ~ C^(2^n) for C = 1.467067423065535412629251121186749718727038915553188083467...
a(n) = 2^(2^(n-1)) * b(n), where b(0) = 0, b(n+1) = b(n)^2 + 1/2. - Jianing Song, Apr 09 2025

A116515 a(n) = the period of the Fibonacci numbers modulo p divided by the smallest m such that p divides Fibonacci(m), where p is the n-th prime.

Original entry on oeis.org

1, 2, 4, 2, 1, 4, 4, 1, 2, 1, 1, 4, 2, 2, 2, 4, 1, 4, 2, 1, 4, 1, 2, 4, 4, 1, 2, 2, 4, 4, 2, 1, 4, 1, 4, 1, 4, 2, 2, 4, 1, 1, 1, 4, 4, 1, 1, 2, 2, 1, 4, 1, 2, 1, 4, 2, 4, 1, 4, 2, 2, 4, 2, 1, 4, 4, 1, 4, 2, 1, 4, 1, 2, 4, 1, 2, 4, 4, 2, 2, 1, 4, 1, 4, 1, 2, 2, 4, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 4, 2, 2, 1

Offset: 1

Nick Krempel, Mar 24 2006

Conditions on p_n mod 4 and mod 5 restrict possible values of a(n). The unknown (?) case is p = 1 mod 4 and (5|p) = 1, equivalently, p = 1 or 9 mod 20, where {1, 2, 4} all occur.

Number of zeros in fundamental period of Fibonacci numbers mod prime(n). [From T. D. Noe, Jan 14 2009]

			a(4) = 2, as 7 is the 4th prime, the Fibonacci numbers mod 7 have period 16, the first Fibonacci number divisible by 7 is F(8) = 21 = 3*7 and 16 / 8 = 2.
One period of the Fibonacci numbers mod 7 is 1, 1, 2, 3, 5, 1, 6, 0, 6, 6, 5, 4, 2, 6, 1, 0, which has two zeros. Hence a(4)=2. [From _T. D. Noe_, Jan 14 2009]

Cf. A060305, A001602.

Cf. A112860, A053027, A053028 (primes producing 1, 2 and 4 zeros) [From T. D. Noe, Jan 14 2009]

a(n) = A060305(n) / A001602(n). a(n) is always one of {1, 2, 4}.

a(n) = A001176(prime(n)) [From T. D. Noe, Jan 14 2009]

A116514 a(1) = 1; thereafter a(n) = (p - (5|p)) divided by the smallest m such that p divides Fibonacci(m), where p is the n-th prime and (5|p) is the Legendre symbol.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 3, 2, 1, 4, 1, 1, 2, 1, 1, 8, 2, 2, 1, 3, 4, 6, 1, 1, 2, 3, 4, 3, 2, 1, 1, 2, 1, 2, 1, 2, 2, 9, 5, 1, 1, 2, 18, 1, 2, 1, 2, 3, 4, 1, 2, 10, 1, 2, 7, 1, 2, 2, 3, 2, 3, 2, 6, 1, 1, 2, 1, 1, 4, 2, 4, 2, 1, 20, 1, 2, 1, 1, 2, 2, 10, 1, 1, 1, 1, 1, 1, 1, 2, 20, 1, 6, 1, 18, 3

Offset: 1

Nick Krempel, Mar 24 2006

Lucas showed that A001602 divides p-1 or p+1, according as (5|p) = 1 or -1 respectively. This is the quotient.

			a(6) = 2, as 13 is the 6th prime, 5 is not a quadratic residue mod 13, 13 first occurs as a prime factor of Fibonacci(7) and (13 - (-1)) / 7 = 2.

Patrick McKinley, Table of n, a(n) for n = 1..10000

Cf. A001602.

a(n) = (prime(n) - (5|prime(n))) / A001602(n).

a(1)=1 added by N. J. A. Sloane, Dec 07 2020

cp's OEIS Frontend

User: Nick Krempel

A332868 Number of involutions (plus identity) in a fixed Sylow 2-subgroup of the symmetric group of degree 2n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A332869 Number of fixed-point free involutions in a fixed Sylow 2-subgroup of the symmetric group of degree 4n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A332840 Number of fixed-point free involutions in a fixed Sylow 2-subgroup of the symmetric group of degree 2n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A332759 Number of involutions (plus identity) in a fixed Sylow 2-subgroup of the symmetric group of degree n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A332757 Number of involutions (plus identity) in the n-fold iterated wreath product of C_2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A332758 Number of fixed-point free involutions in the n-fold iterated wreath product of C_2.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A116515 a(n) = the period of the Fibonacci numbers modulo p divided by the smallest m such that p divides Fibonacci(m), where p is the n-th prime.

Views

Author

Keywords