A001309 -id:A001309 - cp's OEIS frontend

A003956 Order of complex Clifford group of degree 2^n arising in quantum coding theory.

8, 192, 92160, 743178240, 97029351014400, 203286581427673497600, 6819500449352277792129024000, 3660967964237442812098963052691456000, 31446995505814020383166371418359014222725120000

Offset: 0

N. J. A. Sloane and Peter Shor

T. D. Noe, Table of n, a(n) for n = 0..20
Simon Burton, Elijah Durso-Sabina, and Natalie C. Brown, Genons, Double Covers and Fault-tolerant Clifford Gates, arXiv:2406.09951 [quant-ph], 2024. See p. 18.
A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, Quantum error correction via codes over GF(4), arXiv:quant-ph/9608006, 1996-1997; IEEE Trans. Inform. Theory, 44 (1998), 1369-1387.
G. Nebe, E. M. Rains, and N. J. A. Sloane, The invariants of the Clifford groups, arXiv:math/0001038 [math.CO], 2000; Des. Codes Crypt. 24 (2001), 99-121.
G. Nebe, E. M. Rains, and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
Edwin Pednault, An alternative approach to optimal wire cutting without ancilla qubits, arXiv:2303.08287 [quant-ph], 2023.
Tefjol Pllaha, Olav Tirkkonen, and Robert Calderbank, Binary Subspace Chirps, arXiv:2102.12384 [cs.IT], 2021.
Bernhard Runge, Codes and Siegel modular forms, Discrete Math. 148 (1996), 175-204.
Peter Selinger, Generators and relations for n-qubit Clifford operators, arXiv:1310.6813 [quant-ph], 2013; Log. Methods Comput. Sci. 11 (2:10) (2015), 1-17, doi:10.2168/LMCS-11(2:10)2015.
Index entries for sequences related to groups.

Cf. A001309, A014116, A014115, A027672, A100221.

Equals twice A027638.

List([0..10], n-> 2^((n+1)^2 +2)*Product([1..n], j-> 4^j -1) ); # G. C. Greubel, Sep 24 2019

[n eq 0 select 8 else 2^((n+1)^2+2)*(&*[4^j-1: j in [1..n]]): n in [0..10]]; // G. C. Greubel, Sep 24 2019

a(n):= 2^(n^2+2*n+3)*mul(4^j-1, j=1..n); seq(a(n), n=0..10); # modified by G. C. Greubel, Sep 24 2019

Table[2^(n^2+2n+3) Product[4^j-1,{j,n}],{n,0,10}] (* Harvey P. Dale, Nov 03 2017 *)

vector(11, n, 2^(n^2 +2)*prod(j=1,n-1, 4^j-1) ) \\ G. C. Greubel, Sep 24 2019

from math import prod
def A003956(n): return prod((1<Chai Wah Wu, Jun 20 2022

[2^((n+1)^2 +2)*product(4^j -1 for j in (1..n)) for n in (0..10)] # G. C. Greubel, Sep 24 2019

From Amiram Eldar, Jul 06 2025: (Start)
a(n) = 2^(n^2+2*n+3) * Product_{k=1..n} (4^k-1).
a(n) ~ c * 2^(2*n^2+3*n+3), where c = A100221. (End)

A090770 a(n) = 2^(n^2 + 2n + 1)*Product_{j=1..n} (4^j - 1).

Original entry on oeis.org

2, 48, 23040, 185794560, 24257337753600, 50821645356918374400, 1704875112338069448032256000, 915241991059360703024740763172864000, 7861748876453505095791592854589753555681280000, 1080506416218846625176535970968094253434513802154475520000, 2376056471052200653607636735377527394627947719754523173734842368000000

Offset: 0

N. J. A. Sloane, Feb 10 2004

nonn

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). This is the sequence obtained by (illegally) setting p = 2.

G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.

Cf. A001309, A003956, A100221.
Cf. A092299 and A092301 (p=3), A092300 and A089989 (p=5), A090768 and A090769 (p=7).
A bisection of A003053, cf. A003923.

Table[2^(n^2+2n+1) Product[4^j-1,{j,n}],{n,0,10}] (* Harvey P. Dale, May 14 2022 *)

from math import prod
def A090770(n): return prod((1<Chai Wah Wu, Jun 20 2022

a(n) ~ c * 2^((n+1)*(2*n+1)), where c = A100221. - Amiram Eldar, Jul 06 2025

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

N. J. A. Sloane, Feb 10 2004

nonn

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).

G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.

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).

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

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) = 47^(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

N. J. A. Sloane, Feb 10 2004

nonn

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).

G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.

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).

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 *)

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

N. J. A. Sloane, Feb 10 2004

nonn

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).

G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.

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).

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

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) = 43^(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

N. J. A. Sloane, Feb 10 2004

nonn

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).

G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.

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).

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 *)

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) = 25^(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

N. J. A. Sloane, Feb 10 2004

nonn

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).

G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.

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).

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 *)

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

N. J. A. Sloane, Feb 10 2004

nonn

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).

G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.

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).

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

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)

A014115 Order of a certain Clifford group in dimension 2^n (the automorphism group of the Barnes-Wall lattice for n != 3).

Original entry on oeis.org

2, 8, 1152, 2580480, 89181388800, 48126558103142400, 409825748158189771161600, 55428899652335313894424707072000

Offset: 0

N. J. A. Sloane

nonn

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 129.

A. R. Calderbank, E. M. Rains, P. W. Shor and N. J. A. Sloane, Quantum error correction via codes over GF(4), arXiv:quant-ph/9608006, 1996-1997; IEEE Trans. Inform. Theory, 44 (1998), 1369-1387.
G. E. Wall, On the Clifford collineation, transform and similarity groups. IV. An application to quadratic forms, Nagoya Math. J., 21 (1962), pp. 199-222.
Index entries for sequences related to Barnes-Wall lattices.

Agrees with A014116 except at n=3.

Cf. A001309, A003956, A100221.

2^(n^2+n+1) * (2^n - 1) * product('2^(2*i)-1','i'=1..n-1);

a[n_] := 2^(n^2+n+1)*(2^n - 1) * Product[4^i - 1, {i, 1, n-1}]; a[0] = 2; Array[a, 8, 0] (* Amiram Eldar, Jul 07 2025 *)

from math import prod
def A014115(n): return 2 if n == 0 else ((1<Chai Wah Wu, Jun 20 2022

From Amiram Eldar, Jul 07 2025: (Start)
a(n) = 2^(n^2+n+1) * (2^n-1) * Product_{i=1..n-1} (2^(2*i)-1).
a(n) ~ c * 2^(2*n^2+n+1), where c = A100221. (End)

A014116 Order of automorphism group of Barnes-Wall lattice in dimension 2^n.

Original entry on oeis.org

2, 8, 1152, 696729600, 89181388800, 48126558103142400, 409825748158189771161600, 55428899652335313894424707072000

Offset: 0

N. J. A. Sloane

nonn

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 129.

A. R. Calderbank, E. M. Rains, P. W. Shor and N. J. A. Sloane, Quantum error correction via codes over GF(4), arXiv:quant-ph/9608006, 1996-1997; IEEE Trans. Inform. Theory, 44 (1998), 1369-1387.
G. E. Wall, On the Clifford collineation, transform and similarity groups. IV. An application to quadratic forms, Nagoya Math. J., 21 (1962), pp. 199-222.
Index entries for sequences related to Barnes-Wall lattices

Agrees with A014115 except at n=3. Equals half of A001309. Cf. A003956.

2^(n^2+n+1) * (2^n - 1) * product('2^(2*i)-1','i'=1..n-1); # except for n=3.

from math import prod
def A014116(n): return 2+696729598*(n//3) if n == 0 or n == 3 else ((1<Chai Wah Wu, Jun 20 2022

cp's OEIS Frontend

A003956 Order of complex Clifford group of degree 2^n arising in quantum coding theory.

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A090770 a(n) = 2^(n^2 + 2n + 1)*Product_{j=1..n} (4^j - 1).

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

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

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

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

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

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

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A090768 a(n) = 47^(n^2+2n+1)Product_{j=1..n} (49^j-1).

A092299 a(n) = 43^(n^2+2n+1)Product_{j=1..n} (9^j-1).

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