A171499 -id:A171499 - cp's OEIS frontend

A028365 Order of general affine group over GF(2), AGL(n,2).

1, 2, 24, 1344, 322560, 319979520, 1290157424640, 20972799094947840, 1369104324918194995200, 358201502736997192984166400, 375234700595146883504949480652800, 1573079924978208093254925489963584716800

Offset: 0

N. J. A. Sloane

For n > 0, a(n) = v(n+1)/v(n), where v = A203305 is the Vandermonde determinant of the first n of the numbers -2^j - 1; see the Mathematica section. - Clark Kimberling, Jan 01 2012

J. M. Borwein, D. H. Bailey and R. Girgensohn, Experimentation in Mathematics, A K Peters, Ltd., Natick, MA, 2004. x+357 pp. See p. 54 (1.64).

Seiichi Manyama, Table of n, a(n) for n = 0..57
Abdalla G. M. Ahmed, Matt Pharr, Victor Ostromoukhov, and Hui Huang, SZ Sequences: Binary-Based (0, 2^q)-Sequences, arXiv:2505.20434 [cs.GR], 2025. See p. 7.
Marcus Brinkmann, Extended Affine and CCZ Equivalence up to Dimension 4, Ruhr University Bochum (2019).
Putnam Competition 1999, Question A6, Amer. Math. Monthly 107 (Oct 2000), 721-732; see p. 725.
I. Strazdins, Universal affine classification of Boolean functions, Acta Applic. Math. 46 (1997), 147-167.

Cf. A000217, A002884, A005329, A020522, A048651, A053601, A203305.

[1] cat [(&*[2^(n+1) - 2^(j+1): j in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Aug 31 2023

A028365 := n->2^n*product(2^n-2^'i','i'=0..n-1); # version 1
A028365 := n->product(2^'j'-1,'j'=1..n)*2^binomial(n+1,2); # version 2

RecurrenceTable[{a[1]==1, a[2]==2, a[3]==24, a[n]==(6a[n-1]^2 a[n-3] - 8a[n-1] a[n-2]^2)/(a[n-2] a[n-3])}, a[n], {n,20}] (* Harvey P. Dale, Aug 03 2011 *)
(* Next, the connection with Vandermonde determinants *)
f[j_]:= 2^j - 1; z = 15;
v[n_]:= Product[Product[f[k] - f[j], {j,k-1}], {k,2,n}]
Table[v[n], {n,z}]   (* A203303 *)
Table[v[n+1]/v[n], {n,z}]  (* A028365 *)
Table[v[n]*v[n+2]/(2*v[n+1])^2, {n,z}]  (* A171499 *) (* Clark Kimberling, Jan 01 2011 *)
Table[(-1)^n*2^Binomial[n+1,2]*QPochhammer[2,2,n], {n,0,20}] (* G. C. Greubel, Aug 31 2023 *)

a(n)=if(n<0,0,prod(k=1,n,2^k-1)*2^((n^2+n)/2)) /* Michael Somos, May 09 2005 */

[product(2^(n+1) - 2^(k+1) for k in range(n)) for n in range(21)] # G. C. Greubel, Aug 31 2023

a(n) is asymptotic to C*2^(n*(n+1)) where C = 0.288788095086602421278899721... = prod(k>=1, 1-1/2^k) (cf. A048651). - Benoit Cloitre, Apr 11 2003
a(n) = (6*a(n-1)^2*a(n-3) - 8*a(n-1)*a(n-2)^2) / (a(n-2)*a(n-3)). [From Putman Exam]. - Max Alekseyev, May 18 2007
a(n) = 2*A203305(n), n > 0. - Clark Kimberling, Jan 01 2012
From Max Alekseyev, Jun 09 2015: (Start)
a(n) = 2^A000217(n) * A005329(n).
a(n) = 2^n * A002884(n).
a(n) = 2^n * n! * A053601(n). (End)
From G. C. Greubel, Aug 31 2023: (Start)
a(n) = Product_{j=0..n-1} (2^(n+1) - 2^(j+1)).
a(n) = (-1)^n * 2^binomial(n+1,2) * QPochhammer(2,2,n). (End)

A171472 a(n) = 6a(n-1) - 8a(n-2) for n > 1; a(0) = 7, a(1) = 30.

Original entry on oeis.org

7, 30, 124, 504, 2032, 8160, 32704, 130944, 524032, 2096640, 8387584, 33552384, 134213632, 536862720, 2147467264, 8589901824, 34359672832, 137438822400, 549755551744, 2199022731264, 8796091973632, 35184369991680

Offset: 0

Klaus Brockhaus, Dec 09 2009

Related to Reverse and Add trajectory of 22 in base 2: A061561(4*n+2) = 12*a(n).
Third binomial transform of A010729.
a(n) in base 2 is n+3 1s followed by n 0s. - Hussam al-Homsi, Oct 12 2021

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (6,-8).

Cf. A061561, A010729 (repeat 7, 9), A171470, A171471, A171473, A171499.

[8*4^n-2^n: n in [0..30]]; // Vincenzo Librandi, May 31 2011

LinearRecurrence[{6,-8},{7,30},30] (* Harvey P. Dale, Sep 01 2016 *)

{m=22; v=concat([7, 30], vector(m-2)); for(n=3, m, v[n]=6*v[n-1]-8*v[n-2]); v}

a(n) = 8*4^n-2^n.
G.f.: (7-12*x)/((1-2*x)*(1-4*x)).
a(n) = A171499(n+1)/2. - Hussam al-Homsi, Jun 06 2021
E.g.f.: exp(2*x)*(8*exp(2*x) - 1). - Stefano Spezia, Sep 27 2023

A171498 a(n) = 4*3^n-1.

Original entry on oeis.org

3, 11, 35, 107, 323, 971, 2915, 8747, 26243, 78731, 236195, 708587, 2125763, 6377291, 19131875, 57395627, 172186883, 516560651, 1549681955, 4649045867, 13947137603, 41841412811, 125524238435, 376572715307, 1129718145923, 3389154437771, 10167463313315, 30502389939947

Offset: 0

Klaus Brockhaus, Dec 10 2009

Binomial transform of A171497.

Inverse binomial transform of A171499.

Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A010703, A036543, A171497, A171499.

NestList[3#+2&,3,30]  (* Harvey P. Dale, Feb 25 2011 *)

{m=25; v=concat([3], vector(m-1)); for(n=2, m, v[n]=3*v[n-1]+2); v}

a(n) = 3*a(n-1)+2 for n > 0; a(0) = 3.
G.f.: (3-x)/((1-x)*(1-3*x)).
a(n) = A036543(n) + 2. - Philippe Deléham, Apr 13 2013
E.g.f.: exp(x)*(4*exp(2*x) - 1). - Stefano Spezia, Aug 04 2024

a(25)-a(27) from Stefano Spezia, Aug 04 2024

A203303 Vandermonde determinant of the first n terms of (1,2,4,8,16,...).

Original entry on oeis.org

1, 1, 6, 1008, 20321280, 203199794380800, 4096245678214226116608000, 671169825411994707343327912777482240000, 3589459026274030507466469204160461571257625328222208000000, 2511229721141086754031154605327661795863172723306019839389105937236728217600000000

Offset: 1

Clark Kimberling, Jan 01 2012

nonn

Each term divides its successor, as in A002884. Indeed, 2*v(n+1)/v(n) divides v(n+2)/v(n+1), as in A171499.

Robert Israel, Table of n, a(n) for n = 1..22
Daniel M. Kane, Carlo Sanna, and Jeffrey Shallit, Waring's theorem for binary powers, arXiv:1801.04483 [math.NT], Jan 13 2018.

Cf. A000079, A002884, A171499.

[1] cat [(&*[(&*[2^(k+1) -2^j: j in [0..k]]): k in [0..n-2]]): n in [2..15]]; // G. C. Greubel, Aug 31 2023

# First program
with(LinearAlgebra):
a:= n-> Determinant(VandermondeMatrix([2^i$i=0..n-1])):
seq(a(n), n=1..12);  # Alois P. Heinz, Jul 23 2017
# Second program
f:= n -> 2^(n*(n-1)*(n-2)/6)*mul((2^k-1)^(n-k),k=1..n-1):
seq(f(n),n=1..12); # Robert Israel, Jan 16 2018

(* First program *)
f[j_]:= 2^(j-1); z = 15;
v[n_]:= Product[Product[f[k] - f[j], {j,k-1}], {k,2,n}]
Table[v[n], {n,z}]                     (* A203303 *)
Table[v[n+1]/v[n], {n,z}]              (* A002884 *)
Table[v[n]*v[n+2]/(2*v[n+1]^2), {n,z}]  (* A171499 *)
Table[FactorInteger[v[n]], {n,z}]
(* Second program *)
Table[Product[2^(k+1) -2^j, {k,0,n-2}, {j,0,k}], {n,15}] (* G. C. Greubel, Aug 31 2023 *)

[product(product(2^(k+1) -2^j for j in range(k+1)) for k in range(n-1)) for n in range(1,16)] # G. C. Greubel, Aug 31 2023

From Robert Israel, Jan 16 2018: (Start)
a(n) = Product_{0 <= i < j <= n-1} (2^j - 2^i).
a(n) = 2^(n*(n-1)*(n-2)/6) * Product_{1<=k<=n-1} (2^k-1)^(n-k). (End)
a(n) ~ 1/A335011 * 2^(n*(n-1)*(2*n-1)/6) * QPochhammer(1/2)^n. - Vaclav Kotesovec, May 19 2020
a(n) = Product_{k=0..n-2} ( 2^(k+1)^2 * QPochhammer(2^(-k-1); 2; k+1) ). - G. C. Greubel, Aug 31 2023

A130654 Exponent m such that 2^m = A092505(n) = A002430(n) / A046990(n).

Original entry on oeis.org

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

Offset: 1

Alexander Adamchuk, Jun 20 2007, Jun 23 2007

nonn

Conjecture: A092505(n) is always a power of 2. a(n) = Log[ 2, A092505(n) ]. Note that a(n) = 0 iff n is a power of 2; or A002430(2^n) = A046990(2^n) and A092505(2^n) = 1. It appears that a(2k+1) = 1 for k>0. Note that least index k such that a(k) = n is {1, 3, 14, 60, ...} which apparently coincides with A006502(n) = {1, 3, 14, 60, 279, 1251, ...} Related to Fibonacci numbers (see Carlitz reference).
Least index k such that a(k) = n is listed in A131262(n) = {1, 3, 14, 60, 248, ...}. Conjecture: A131262(n) = Sigma(2^n)*EulerPhi(2^n) = 2^(2n) - Floor(2^n/2) = A062354(2^n). If this conjecture is true then a(1008) = 5 and a(n)<5 for all n<1008.
Positions of records indeed continue as 1, 3, 14, 60, 248, 1008, 4064, 16320, ..., strongly suggesting union of {1} and A171499. - Antti Karttunen, Jan 13 2019

			A092505(n) begins {1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 4, 2, 1, 2, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 4, 2, 4, 2, 1, ...}.
Thus a(1) = Log[2,1] = 0, a(2) = Log[2,1] = 0, a(3) = Log[2,2] = 1.

Antti Karttunen, Table of n, a(n) for n = 1..16385
L. Carlitz, Some polynomials related to Fibonacci and Eulerian numbers, Fib. Quart., 16 (1978), 216-226.

Cf. A092505 = A002430(n) / A046990(n), n>0. Cf. A002430 = Numerators in Taylor series for tan(x). Cf. A046990 = Numerators of Taylor series for log(1/cos(x)). Cf. A006502 = Related to Fibonacci numbers.
Cf. A131262 = Least index k such that A130654(k) = n. Cf. A062354 = Sigma(n)*EulerPhi(n).
Cf. also A171499.

a=Series[ Tan[x], {x,0,256} ]; b=Series[ Log[ 1/Cos[x] ], {x,0,256}]; Table[ Log[ 2, Numerator[ SeriesCoefficient[ a, 2n-1 ] ] / Numerator[ SeriesCoefficient[ b, 2n ] ] ], {n,1,128} ]

a(n) = Log[ 2, A092505(n) ]. a(n) = Log[ 2, A002430(n) / A046990(n) ] = A007814(A092505(n)).

A171496 a(n) = 6a(n-1) - 8a(n-2) for n > 1; a(0) = 6, a(1) = 28.

Original entry on oeis.org

6, 28, 120, 496, 2016, 8128, 32640, 130816, 523776, 2096128, 8386560, 33550336, 134209536, 536854528, 2147450880, 8589869056, 34359607296, 137438691328, 549755289600, 2199022206976, 8796090925056, 35184367894528

Offset: 0

Klaus Brockhaus, Dec 10 2009

Binomial transform of A171495; second binomial transform of A171494; third binomial transform of A010726.

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (6,-8).

Equals 2*A171499.

Cf. A171476, A006516, A010036, A171494, A171495, A010726, A171472, A171473.

[8*4^n-2*2^n: n in [0..30]]; // Vincenzo Librandi, Jul 18 2011

LinearRecurrence[{6,-8},{6,28},30] (* Harvey P. Dale, Dec 21 2014 *)

{m=22; v=concat([6, 28], vector(m-2)); for(n=3, m, v[n]=6*v[n-1]-8*v[n-2]); v}

a(n) = 8*4^n - 2*2^n.
G.f.: 2*(3-4*x)/((1-2*x)*(1-4*x)).
a(n) = A171476(n+1) = A006516(n+2).
a(n+1) - a(n) = A010036(n+2).
a(n) = 4*a(n-1)+2^(n+1) (with a(0)=6). - Vincenzo Librandi, Dec 04 2010
E.g.f.: 2*exp(2*x)*(2*exp(2*x) - 1)*(2*exp(2*x) + 1). - Stefano Spezia, Dec 10 2021

A266214 Numbers n that are not coprime to the numerator of zeta(2n)/(Pi^(2n)).

Original entry on oeis.org

14, 22, 26, 28, 30, 38, 42, 44, 46, 50, 52, 54, 56, 58, 60, 62, 70, 74, 76, 78, 82, 84, 86, 88, 90, 92, 94, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 134, 138, 140, 142, 146, 148, 150, 152, 154, 156, 158, 162, 164, 166, 168, 170

Offset: 1

Chris Boyd, Robert Israel, Dec 24 2015

Equivalently, n not coprime to the numerator of 2^(2n-1)*Bernoulli(2n)/(2n)! (see Lekraj Beedassy comment in A046988).
Conjecture 1: for n>=1, a(n) is identical to 2*A072823(n+1).
Conjecture 2: The corresponding GCDs are powers of 2.
Verified for n <= 10000, e.g.,
GCD = 2 for 14, 22, 26, 28, 30, 38, 42, 44, 46, 50, 52, 54, 56, 58, ...
GCD = 4 for 60, 92, 108, 116, 120, 124, 156, 172, 180, 184, 188, ...
GCD = 8 for 248, 376, 440, 472, 488, 496, 504, 632, 696, 728, 744, ...
GCD = 16 for 1008, 1520, 1776, 1904, 1968, 2000, 2016, 2032, 2544, ...
GCD = 32 for 4064, 6112, 7136, 7648, 7904, 8032, 8096, 8128, 8160
Taking GCDs vertically, column 1 = "14, 60, 248, 1008, 4064, ..." appears to be essentially the same as A171499 and A131262; (ii) column 2 = "22, 92, 376, 1520, 6112, ..." appears to be essentially the same as A010036.
From Chris Boyd, Jan 25 2016: (Start)
Determining whether n is a term of this sequence can be approached by considering odd and even factors separately, and exploiting the fact that numerator(zeta(2n)/(Pi^(2n))) = numerator(2^(2n-2)*N_2n/(D_2n*(2n)!)), where N_2n and D_2n are odd coprime integers such that Bernoulli(2n) = N_2n/2D_2n.
Case 1: odd factors. n is a term if it has an odd prime factor p (necessarily irregular) that divides N_2n at a higher multiplicity than it divides (2n)!. No such factor p of N_2n up to 2n = 10000 is of sufficient multiplicity, and the apparent scarcity of squared and higher power factors of N_2n values (see A090997) suggests that no such p is likely to exist.
Case 2: even factors. An even n is a term if 2 divides 2^(2n-2) at a higher multiplicity than it divides (2n)!. The multiplicity of 2 in 2^(2n-2) is 2n-2, and in (2n)! is 2n minus the number of 1's in the binary expansion of 2n (see A005187). Qualifying n values are therefore those where the number of 1's in the binary expansion of 2n is greater than 2. Except for its first term, A072823 comprises integers with three or more 1-bits in their binary expansion. It follows that for m > 1, 2*A072823(m) values belong to this sequence.
In summary, this sequence is essentially a supersequence of 2*A072823. Conjectures 1 and 2 are true if there are no irregular odd primes p that divide n and the numerator of Bernoulli(2n)/(2n)!. (End)

Chris Boyd, Table of n, a(n) for n = 1..10000
C. Boyd in reply to R. Israel, A046988 query, SeqFan list, Dec 22 2015.

Cf. A010036, A072823, A131262, A171499.

select(n -> igcd(n,numer(2^(2*n-1)*bernoulli(2*n)/(2*n)!)) > 1), [$1..1000]);

Select[Range@ 170, ! CoprimeQ[#, Numerator[Zeta[2 #]/Pi^(2 #)]] &] (* Michael De Vlieger, Dec 24 2015 *)

test(n) = if(gcd(numerator(2^(2*n-1)*bernfrac(2*n)/(2*n)!),n)!=1,1,0)
for(i=1,200,if(test(i),print1(i,", ")))

A187560 a(n) = 4^(n+1)-2^n-1.

Original entry on oeis.org

2, 13, 59, 247, 1007, 4063, 16319, 65407, 261887, 1048063, 4193279, 16775167, 67104767, 268427263, 1073725439, 4294934527, 17179803647, 68719345663, 274877644799, 1099511103487, 4398045462527, 17592183947263, 70368739983359, 281474968322047, 1125899890065407

Offset: 0

Brad Clardy, Mar 25 2011

For n>0, binary numbers of the form (n+1)0 n, where n is the index value and the number of 1's. This can be formed by appending a leading 1 to the terms of A129868. It is also A156589 written in bit-reverse order.

			Binary values of the first 7 terms are 10, 1101, 111011, 11110111, 1111101111, 111111011111, 11111110111111.

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Cf. A171499.

Table[4^(n+1)-2^n-1,{n,0,30}] (* or *) LinearRecurrence[{7,-14,8},{2,13,59},30] (* Harvey P. Dale, Feb 25 2013 *)

a(n)=4^(n+1)-2^n-1 \\ Charles R Greathouse IV, Nov 01 2015

a(n) = 4^(n+1)-2^n-1 = A171499(n)-1.
G.f.: ( -2+x+4*x^2 ) / ( (x-1)*(2*x-1)*(4*x-1) ). - R. J. Mathar, Apr 09 2011
a(0)=2, a(1)=13, a(2)=59, a(n)=7*a(n-1)-14*a(n-2)+8*a(n-3). - Harvey P. Dale, Feb 25 2013
E.g.f.: exp(x)*(4*exp(3*x) - exp(x) - 1). - Stefano Spezia, Apr 11 2025

Terms a(21) and beyond from Andrew Howroyd, Feb 25 2018

cp's OEIS Frontend

A028365 Order of general affine group over GF(2), AGL(n,2).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

SageMath

Formula

A171472 a(n) = 6*a(n-1) - 8*a(n-2) for n > 1; a(0) = 7, a(1) = 30.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A171498 a(n) = 4*3^n-1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A203303 Vandermonde determinant of the first n terms of (1,2,4,8,16,...).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

SageMath

Formula

A130654 Exponent m such that 2^m = A092505(n) = A002430(n) / A046990(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A171496 a(n) = 6*a(n-1) - 8*a(n-2) for n > 1; a(0) = 6, a(1) = 28.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A266214 Numbers n that are not coprime to the numerator of zeta(2*n)/(Pi^(2*n)).

Views

Author

Keywords

A171472 a(n) = 6a(n-1) - 8a(n-2) for n > 1; a(0) = 7, a(1) = 30.

A171496 a(n) = 6a(n-1) - 8a(n-2) for n > 1; a(0) = 6, a(1) = 28.

A266214 Numbers n that are not coprime to the numerator of zeta(2n)/(Pi^(2n)).