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 1-7 of 7 results.

A003053 Order of orthogonal group O(n, GF(2)).

Original entry on oeis.org

1, 2, 6, 48, 720, 23040, 1451520, 185794560, 47377612800, 24257337753600, 24815256521932800, 50821645356918374400, 208114637736580743168000, 1704875112338069448032256000, 27930968965434591767112450048000, 915241991059360703024740763172864000
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • 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].
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Bisections give A003923 and A090770.

Programs

  • Maple
    h:=proc(n) local m;
if n mod 2 = 0 then m:=n/2;
2^(m^2)*mul( 4^i-1, i=1..m);
else m:=(n+1)/2;
2^(m^2)*mul( 4^i-1, i=1..m-1);
fi;
end;
# This produces a(n+1)
  • Mathematica
    h[n_] := Module[{m}, If[EvenQ[n], m = n/2; 2^(m^2)*Product[4^i-1, {i, 1, m}], m = (n+1)/2; 2^(m^2)*Product[4^i-1, {i, 1, m-1}]]];
a[n_] := h[n-1];
Array[a, 16] (* Jean-François Alcover, Aug 18 2022, after Maple code *)
  • PARI
    a(n) = n--; if (n % 2, m = (n+1)/2; 2^(m^2)*prod(k=1, m-1, 4^k-1), m = n/2; 2^(m^2)*prod(k=1, m, 4^k-1)); \\ Michel Marcus, Jul 13 2017
  • Python
    def size_binary_orthogonal_group(n):
    k = n-1
    if k%2==0:
        m=k//2
        p=2**(m**2)
        for i in range(1,m+1):
            p*=4**i-1
    else:
        m=(k+1)//2
        p=2**(m**2)
        for i in range(1,m):
            p*=4**i-1
    return p
#call and print output for a(n)
print([size_binary_orthogonal_group(n) for n in range(1, 10)])
# Nathan J. Russell, Nov 01 2017
  • Python
    from math import prod
def A003053(n): return (1 << (n//2)**2)*prod((1 << i)-1 for i in range(2,2*((n-1)//2)+1,2)) # Chai Wah Wu, Jun 20 2022

Formula

For formulas see Maple code.
Asymptotics: a(n) ~ c * 2^((n^2-n)/2), where c = (1/4; 1/4)infinity ~ 0.6885375... is expressed in terms of the Q-Pochhammer symbol. - _Cedric Lorand, Aug 07 2017

Extensions

Edited by N. J. A. Sloane, Dec 30 2008
Edited by W. Edwin Clark et al., Jan 15 2015

A089989 a(n) = 5^(n^2+2n+1)*Product_{j=1..n} (25^j-1).

Original entry on oeis.org

5, 15000, 29250000000, 35703281250000000000, 27239372138671875000000000000000, 12988743471794208526611328125000000000000000000, 3870947187719439049405530095100402832031250000000000000000000000, 721020100095350865678782984846420731628313660621643066406250000000000000000000000000
Offset: 0

Views

Author

N. J. A. Sloane, Feb 10 2004

Keywords

Comments

The order of the p-Clifford group for an odd prime p is a*p^(n^2+2n+1)*Product_{j=1..n} (p^(2*j)-1), where a = gcd(p+1,4).

Links

Crossrefs

Cf. A001309, A003956.
Cf. A092299 and A092301 (p=3), A092300 and A089989 (p=5), A090768 and A090769 (p=7), A090770 (p=2, although this is the wrong formula in that case).

Programs

  • Mathematica
    a[n_] := 5^(n^2+2*n+1) * Product[25^j - 1, {j, 1, n}]; Array[a, 10, 0] (* Amiram Eldar, Jul 07 2025 *)

Formula

From Amiram Eldar, Jul 07 2025: (Start)
a(n) = A092300(n) / 2.
a(n) ~ c * 5^(2*n^2+3*n+1), where c = Product_{k>=1} (1 - 1/5^(2*k)) = 0.958400102563... . (End)

A090768 a(n) = 4*7^(n^2+2n+1)*Product_{j=1..n} (49^j-1).

Original entry on oeis.org

28, 460992, 18594942105600, 1801630225452634420838400, 419114092659655895262507217606410240000, 234094442205343557204838431982679810784254737891983360000, 313936710456644712932526713436974934772339799367593873556694922893983744000000, 1010846620958915523772074873493863525346718205399610275113597795065777917926818948851860049494016000000
Offset: 0

Views

Author

N. J. A. Sloane, Feb 10 2004

Keywords

Comments

The order of the p-Clifford group for an odd prime p is a*p^(n^2+2n+1)*Product_{j=1..n} (p^(2*j)-1), where a = gcd(p+1,4).

Links

Crossrefs

Cf. A001309, A003956.
Cf. A092299 and A092301 (p=3), A092300 and A089989 (p=5), A090768 and A090769 (p=7), A090770 (p=2, although this is the wrong formula in that case).

Programs

  • Mathematica
    a[n_] := 4*7^(n^2+2*n+1) * Product[49^j - 1, {j, 1, n}]; Array[a, 10, 0] (* Amiram Eldar, Jul 07 2025 *)

Formula

From Amiram Eldar, Jul 07 2025: (Start)
a(n) = 4 * A090769(n).
a(n) ~ c * 7^(2*n^2+3*n+1), where c = 4 * Product_{k>=1} (1 - 1/7^(2*k)) = 3.916701388593... . (End)

A090769 a(n) = 7^(n^2+2n+1)*Product_{j=1..n} (49^j-1).

Original entry on oeis.org

7, 115248, 4648735526400, 450407556363158605209600, 104778523164913973815626804401602560000, 58523610551335889301209607995669952696063684472995840000, 78484177614161178233131678359243733693084949841898468389173730723495936000000, 252711655239728880943018718373465881336679551349902568778399448766444479481704737212965012373504000000
Offset: 0

Views

Author

N. J. A. Sloane, Feb 10 2004

Keywords

Comments

The order of the p-Clifford group for an odd prime p is a*p^(n^2+2n+1)*Product_{j=1..n} (p^(2*j)-1), where a = gcd(p+1,4).

Links

Crossrefs

Cf. A001309, A003956.
Cf. A092299 and A092301 (p=3), A092300 and A089989 (p=5), A090768 and A090769 (p=7), A090770 (p=2, although this is the wrong formula in that case).

Programs

  • Mathematica
    Table[7^(n^2 + 2 n + 1)*Product[49^j - 1, {j, n}], {n, 0, 7}] (* Wesley Ivan Hurt, Oct 15 2023 *)

Formula

From Amiram Eldar, Jul 07 2025: (Start)
a(n) = A090768(n) / 4.
a(n) ~ c * 7^(2*n^2+3*n+1), where c = Product_{k>=1} (1 - 1/7^(2*k)) = 0.979175347148... . (End)

A092299 a(n) = 4*3^(n^2+2n+1)*Product_{j=1..n} (9^j-1).

Original entry on oeis.org

12, 2592, 50388480, 80225312993280, 10358730921842550374400, 108354149159204252828272715366400, 91807063616969429053277006948134413139968000, 6300752103463414524173850924959140409591369032708128768000, 35026261744325078751960598643637064012678383486922588643915999981076480000
Offset: 0

Views

Author

N. J. A. Sloane, Feb 10 2004

Keywords

Comments

The order of the p-Clifford group for an odd prime p is a*p^(n^2+2n+1)*Product_{j=1..n} (p^(2*j)-1), where a = gcd(p+1,4).

Links

Crossrefs

Cf. A001309, A003956, A132037.
Cf. A092299 and A092301 (p=3), A092300 and A089989 (p=5), A090768 and A090769 (p=7), A090770 (p=2, although this is the wrong formula in that case).

Programs

  • Mathematica
    a[n_] := 4*3^(n^2+2*n+1) * Product[9^j - 1, {j, 1, n}]; Array[a, 10, 0] (* Amiram Eldar, Jul 06 2025 *)

Formula

From Amiram Eldar, Jul 06 2025: (Start)
a(n) = 4 * A092301(n).
a(n) ~ c * 3^(2*n^2+3*n+1), where c = 4 * A132037. (End)

A092300 a(n) = 2*5^(n^2+2n+1)*Product_{j=1..n} (25^j-1).

Original entry on oeis.org

10, 30000, 58500000000, 71406562500000000000, 54478744277343750000000000000000, 25977486943588417053222656250000000000000000000, 7741894375438878098811060190200805664062500000000000000000000000, 1442040200190701731357565969692841463256627321243286132812500000000000000000000000000
Offset: 0

Views

Author

N. J. A. Sloane, Feb 10 2004

Keywords

Comments

The order of the p-Clifford group for an odd prime p is a*p^(n^2+2n+1)*Product_{j=1..n} (p^(2*j)-1), where a = gcd(p+1,4).

Links

Crossrefs

Cf. A001309, A003956.
Cf. A092299 and A092301 (p=3), A092300 and A089989 (p=5), A090768 and A090769 (p=7), A090770 (p=2, although this is the wrong formula in that case).

Programs

  • Mathematica
    a[n_] := 2*5^(n^2+2*n+1) * Product[25^j - 1, {j, 1, n}]; Array[a, 10, 0] (* Amiram Eldar, Jul 07 2025 *)

Formula

From Amiram Eldar, Jul 07 2025: (Start)
a(n) = 2 * A089989(n).
a(n) ~ c * 5^(2*n^2+3*n+1), where c = 2 * Product_{k>=1} (1 - 1/5^(2*k)) = 1.916800205127... . (End)

A092301 a(n) = 3^(n^2+2n+1)*Product_{j=1..n} (9^j-1).

Original entry on oeis.org

3, 648, 12597120, 20056328248320, 2589682730460637593600, 27088537289801063207068178841600, 22951765904242357263319251737033603284992000, 1575188025865853631043462731239785102397842258177032192000, 8756565436081269687990149660909266003169595871730647160978999995269120000
Offset: 0

Views

Author

N. J. A. Sloane, Feb 10 2004

Keywords

Comments

The order of the p-Clifford group for an odd prime p is a*p^(n^2+2n+1)*Product_{j=1..n} (p^(2*j)-1), where a = gcd(p+1,4).

Links

Crossrefs

Cf. A001309, A003956, A132037.
Cf. A092299 and A092301 (p=3), A092300 and A089989 (p=5), A090768 and A090769 (p=7), A090770 (p=2, although this is the wrong formula in that case).

Programs

  • Mathematica
    Table[3^(n^2+2n+1) Product[9^j-1,{j,n}],{n,0,10}] (* Harvey P. Dale, Jun 23 2013 *)

Formula

From Amiram Eldar, Jul 07 2025: (Start)
a(n) = A092299(n) / 4.
a(n) ~ c * 3^(2*n^2+3*n+1), where c = A132037. (End)
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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