A100221 -id:A100221 - cp's OEIS frontend

A371749 Decimal expansion of Product_{k>=0} 1 / (1 + 1/4^k).

3, 6, 8, 7, 5, 6, 1, 2, 7, 0, 7, 6, 9, 0, 0, 5, 6, 2, 7, 5, 0, 8, 4, 5, 6, 7, 2, 2, 8, 0, 8, 1, 9, 9, 1, 5, 4, 8, 2, 3, 4, 5, 1, 7, 9, 9, 3, 7, 7, 2, 5, 5, 6, 2, 1, 4, 5, 7, 1, 2, 5, 1, 8, 3, 4, 8, 2, 0, 1, 5, 2, 5, 6, 3, 3, 8, 8, 4, 1, 8, 4, 1, 2, 8, 9, 5, 9, 5, 4, 3, 5, 4, 4, 3, 4, 5, 0, 1, 1, 3

Offset: 0

Ilya Gutkovskiy, Apr 05 2024

			0.368756127076900562750845672280819915482...

Cf. A000302, A083864, A100221, A273413, A371746, A371747, A371748, A371752.

RealDigits[1/QPochhammer[-1, 1/4], 10, 100][[1]]

Equals A273413^2. - Hugo Pfoertner, Apr 05 2024

A001308 Order of Chevalley group D_n (2).

Original entry on oeis.org

1, 36, 20160, 174182400, 23499295948800, 50027557148216524800, 1691555775522928280469504000, 911666827031785075278550369566720000, 7846393898179181843651374899795632943267840000

Offset: 1

N. J. A. Sloane

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites], p. xvi.
H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 131.

T. D. Noe, Table of n, a(n) for n = 1..20
Index entries for sequences related to groups.

Cf. A003923, A100221.

for m from 0 to 25 do N:=2^(m*(m-1))*mul( 4^i-1, i=1..m-1)*(2^m-1); lprint(N); od:

d[q_, n_] := q^(n*(n-1)) * (q^n-1) * Product[q^(2*k) - 1, {k, 1, n-1}]; Table[d[2, n], {n, 1, 9}] (* Amiram Eldar, Jul 07 2025 *)

a(n) = 2^(n*(n-1)) * (2^n - 1) * Product_{i=1..n-1} (2^(2*i) - 1).

a(n) ~ c * 2^(n*(2*n-1)), where c = A100221. - Amiram Eldar, Jul 07 2025

A028666 a(n) = order of the orthogonal group O_n(2) if n is odd or O^(+)_n(2) if n is even.

Original entry on oeis.org

1, 12, 2880, 11612160, 758041804800, 794088208701849600, 13319336815141167562752000, 3575164027575627746190393606144000, 15354978274323252140217954794120612413440000, 1055182047088717407398960909148529544369642384916480000, 1160183823755957350394353874696058298158177597536388268425216000000

Offset: 0

Olivier Gérard

nonn

Pseudo-Galois numbers for d=4; order of group AGL(n,2^2).

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites], p. xii (but beware typos!).

Vincenzo Librandi, Table of n, a(n) for n = 0..39

Bisections give A144546 and A144547.
Cf. A028668, A028670, A028671, A028672, A028674, A028676, A028677, A028678, A028680, A028682, A028683, A028684, A028686.
Cf. A100221.

f:=proc(n,eps) local m,d;
if n mod 2 = 0 then m:=n/2; d:=gcd(4,2^m-eps);
2^(m*(m-1))*mul( 4^i-1, i=1..m)*(2^m-eps)/d;
else m:=(n-1)/2;
2^(m^2)*mul( 4^i-1, i=1..m);
fi; end;
[seq(f(n,+1),n=0..20)]

FoldList[ #1*4^#2 (4^#2-1)&, 1, Range[ 20 ] ]
a[n_] := 4^n * Product[4^n - 4^k, {k, 0, n-1}]; Array[a, 10, 0] (* Amiram Eldar, Jul 14 2025 *)

a(n) = 4^n * prod(k = 0, n-1, 4^n - 4^k); \\ Amiram Eldar, Jul 14 2025

a(n) = 4^n * Product_{k=0..n-1} (4^n - 4^k).

a(n) ~ c * 4^(n^2+n), where c = A100221. - Amiram Eldar, Jul 14 2025

Entry revised by N. J. A. Sloane, Dec 30 2008

Duplicate term 1 removed by Amiram Eldar, Jul 14 2025

A330862 Decimal expansion of Product_{k>=1} (1 - 1/(-2)^k).

Original entry on oeis.org

1, 2, 1, 0, 7, 2, 4, 1, 3, 0, 3, 0, 1, 0, 5, 9, 1, 8, 0, 1, 3, 6, 1, 7, 2, 8, 5, 6, 1, 0, 5, 9, 0, 5, 0, 4, 6, 3, 6, 8, 0, 4, 1, 6, 3, 1, 1, 2, 3, 1, 3, 7, 6, 4, 3, 4, 7, 6, 1, 5, 9, 2, 4, 5, 5, 4, 0, 0, 0, 6, 8, 7, 5, 6, 5, 9, 1, 8, 4, 5, 0, 4, 9, 9, 1, 6, 5, 0, 7, 6, 1, 0, 1, 3, 3, 5, 5, 5, 3, 9, 5, 3, 9, 9, 6, 4, 6, 3, 3, 0, 9

Offset: 1

Ilya Gutkovskiy, Apr 28 2020

			(1 + 1/2) * (1 - 1/2^2) * (1 + 1/2^3) * (1 - 1/2^4) * (1 + 1/2^5) * ... = 1.2107241303010591801361728561...

Cf. A000203, A048651, A065446, A079555, A081845, A085011, A100221, A132020, A330863.

RealDigits[QPochhammer[-1/2, -1/2], 10, 111] [[1]]
N[QPochhammer[-2, 1/4]*QPochhammer[1/4]/3, 120] (* Vaclav Kotesovec, Apr 28 2020 *)

prodinf(k=1, 1 - 1/(-2)^k) \\ Michel Marcus, Apr 28 2020

Equals Product_{k>=1} (4^k - 1)*(4^k + 2)/4^(2*k).

Equals exp(-Sum_{k>=1} A000203(k)/(k*(-2)^k)).

A371747 Decimal expansion of Product_{k>=1} 1 / (1 - 1/4^k).

Original entry on oeis.org

1, 4, 5, 2, 3, 5, 3, 6, 4, 2, 4, 4, 9, 5, 9, 7, 0, 1, 5, 8, 3, 4, 7, 1, 3, 0, 2, 2, 4, 8, 5, 2, 7, 4, 8, 7, 3, 3, 6, 1, 2, 2, 7, 9, 7, 8, 8, 0, 7, 9, 2, 6, 3, 4, 9, 6, 2, 5, 3, 8, 2, 7, 1, 8, 4, 0, 3, 6, 8, 6, 3, 0, 0, 7, 8, 1, 9, 4, 0, 3, 1, 6, 1, 0, 1, 4, 0, 7, 2, 0, 6, 0, 1, 5, 3, 6, 2, 1, 2, 9

Offset: 1

Ilya Gutkovskiy, Apr 05 2024

			1.4523536424495970158347130224852748733612...

Kelvin Voskuijl, Table of n, a(n) for n = 1..20000

Cf. A000302, A065446, A100221, A370466, A371748, A371749, A371750.

RealDigits[1/QPochhammer[1/4, 1/4], 10, 100][[1]]

Equals 1 / A100221.

A371748 Decimal expansion of Product_{k>=0} (1 + 1/4^k).

Original entry on oeis.org

2, 7, 1, 1, 8, 1, 9, 3, 4, 7, 7, 2, 6, 9, 5, 8, 7, 6, 0, 6, 9, 1, 0, 8, 8, 4, 6, 9, 7, 0, 7, 9, 1, 8, 6, 0, 2, 4, 4, 3, 3, 9, 9, 0, 8, 5, 6, 7, 4, 8, 8, 5, 4, 9, 4, 6, 9, 3, 0, 8, 0, 6, 2, 9, 0, 0, 6, 0, 2, 6, 2, 3, 6, 1, 3, 0, 5, 9, 7, 7, 8, 0, 0, 9, 7, 8, 7, 7, 4, 0, 5, 2, 5, 2, 1, 4, 6, 0, 4, 6

Offset: 1

Ilya Gutkovskiy, Apr 05 2024

			2.71181934772695876069108846970791860244...

Kelvin Voskuijl, Table of n, a(n) for n = 1..20000

Cf. A000302, A065445, A081845, A100221, A132323, A371747, A371749, A371751.

RealDigits[QPochhammer[-1, 1/4], 10, 100][[1]]

Equals A065445^2. - Hugo Pfoertner, Apr 05 2024

A092418 A sieve: starting with the sequence of positive integers, delete every 4th number, then delete every 16th number from the remaining sequence, then delete every 64th number, etc. Sequence gives the remaining numbers.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 22, 23, 25, 26, 27, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 43, 45, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83, 86, 87, 89, 91, 93, 94, 95, 97, 98, 99, 101, 102

Offset: 1

Charles T. Le (charlestle(AT)yahoo.com), Mar 22 2004

The asymptotic density of this sequence is Product_{k>=1} (1 - 1/4^k) = 0.688537... (A100221). - Amiram Eldar, Mar 21 2021

C. Dumitrescu & V. Seleacu, editors, Some Notions and Questions in Number Theory, Vol. I, Erhus Publ., Glendale, 1994.
Florentin Smarandache, Properties of Numbers, 1972.

Amiram Eldar, Table of n, a(n) for n = 1..10000
C. Dumitrescu & V. Seleacu, editors, Some Notions and Questions in Number Theory, Vol. I, item 31.

Cf. A007950, A007951, A100221.

A = 1:200; A(4:4:end) = 0; A = A(find(A)); A(16:16:end) = 0; A = A(find(A)); A(64:64:end) = 0; A = A(find(A))
% David Wasserman, Apr 28 2004

Edited by David Wasserman, Apr 28 2004

A144546 A bisection of A028666.

Original entry on oeis.org

12, 11612160, 794088208701849600, 3575164027575627746190393606144000, 1055182047088717407398960909148529544369642384916480000, 20410164807073092317242309800149338693366138889849970301267088483593224192000000, 25872955740757748502626229629361173659982454517929458713719920139287952355803151825297413315474342543360000000

Offset: 0

N. J. A. Sloane, Dec 30 2008

nonn

Cf. A028666, A100221, A144547.

a[n_] := 4^(2*n+1) * Product[4^(2*n+1) - 4^k, {k, 0, 2*n}]; Array[a, 7, 0] (* Amiram Eldar, Jul 14 2025 *)

a(n) = 4^(2*n+1) * prod(k = 0, 2*n, 4^(2*n+1) - 4^k); \\ Amiram Eldar, Jul 14 2025

From Amiram Eldar, Jul 14 2025: (Start)
a(n) = A028666(2*n+1).
a(n) ~ c * 16^(2*n^2+3*n+1), where c = A100221. (End)

a(0) = 1 removed by Amiram Eldar, Jul 14 2025

A144547 A bisection of A028666.

Original entry on oeis.org

1, 2880, 758041804800, 13319336815141167562752000, 15354978274323252140217954794120612413440000, 1160183823755957350394353874696058298158177597536388268425216000000, 5744950321305805807513301436668994962443746225944514592041927656983526026246267317780480000000, 1864342934383580231084517260259192252139946430124547822192277172067265954040486594733297802503864086641393952539593932800000000

Offset: 0

N. J. A. Sloane, Dec 30 2008

nonn

Cf. A028666, A100221, A144546.

a[n_] := 16^n * Product[16^n - 4^k, {k, 0, 2*n-1}]; Array[a, 8, 0] (* Amiram Eldar, Jul 14 2025 *)

a(n) = 16^n * prod(k = 0, 2*n-1, 16^n - 4^k); \\ Amiram Eldar, Jul 14 2025

From Amiram Eldar, Jul 14 2025: (Start)
a(n) = A028666(2*n).
a(n) ~ c * 16^(2*n^2+n), where c = A100221. (End)

A382979 a(n) = [(x*y)^n] Product_{k>=1} 1/(1 - x^k + y^k).

Original entry on oeis.org

1, -2, 4, -20, 78, -282, 1048, -4014, 15456, -59224, 227646, -879694, 3407730, -13219372, 51375286, -200021556, 779870542, -3044448644, 11898709560, -46553635346, 182315752476, -714619687038, 2803342734160, -11005274516610, 43233909672938, -169951684067602, 668474115081988

Offset: 0

Seiichi Manyama, Apr 11 2025

sign

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Main diagonal of A382974.

Cf. A322211, A100221.

nmax := 26; prec := 2*nmax + 10; Rx := PowerSeriesRing(Rationals(), prec); Rxy := PowerSeriesRing(Rx, prec); P := Rxy!1; for k in [1..prec] do P *:= 1/(1 - x^k + y^k); end for; seq := [Coefficient(Coefficient(P, n), n) : n in [0..nmax]]; print seq; // Vincenzo Librandi, Apr 12 2025

a[n_]:=SeriesCoefficient[Product[1/(1-x^k+y^k),{k,1,n+5}],{x,0,n},{y,0,n}]; Table[a[n],{n,0,26}] (* Vincenzo Librandi, Apr 12 2025 *)

a(n) ~ (-1)^n * 4^n / (A100221 * sqrt(Pi*n)). - Vaclav Kotesovec, Apr 13 2025

cp's OEIS Frontend

A371749 Decimal expansion of Product_{k>=0} 1 / (1 + 1/4^k).

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A001308 Order of Chevalley group D_n (2).

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A028666 a(n) = order of the orthogonal group O_n(2) if n is odd or O^(+)_n(2) if n is even.

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A330862 Decimal expansion of Product_{k>=1} (1 - 1/(-2)^k).

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A371747 Decimal expansion of Product_{k>=1} 1 / (1 - 1/4^k).

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A371748 Decimal expansion of Product_{k>=0} (1 + 1/4^k).

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Programs

Mathematica

Formula

A092418 A sieve: starting with the sequence of positive integers, delete every 4th number, then delete every 16th number from the remaining sequence, then delete every 64th number, etc. Sequence gives the remaining numbers.

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A144546 A bisection of A028666.

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