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 1-5 of 5 results.

A074000 Number of elements of GF(3^n) with trace 0 and subtrace 0.

Original entry on oeis.org

1, 1, 3, 9, 21, 99, 225, 729, 2187, 6561, 19845, 58563, 177633, 531441, 1594323, 4782969, 14344533, 43059843, 129127041, 387420489, 1162261467
Offset: 1

Views

Author

Frank Ruskey and Nate Kube, Aug 19 2002

Keywords

Links

Crossrefs

Cf. A074001, A074002, A074003, A074004, A074005.
Cf. A073947, A053548.

Programs

  • Sage
    def a(n):
    if n==1: return 1
    ans = 0
    for x in GF(3^n):
        if x.charpoly().coefficients(sparse=False)[-3:-1]==[0, 0]: ans += 1
    return ans  # Robin Visser, Dec 28 2024

Formula

Appears to satisfy a linear recurrence of order 5 with signature (0, 3, 9, 18, 27). This also applies to sequences A074001-A074005. - Andrey Zabolotskiy, Dec 30 2024

Extensions

a(18) corrected and a(21) added by Robin Visser, Dec 28 2024

A074005 Number of elements of GF(3^n) with trace 1 and subtrace 2.

Original entry on oeis.org

0, 2, 3, 6, 30, 81, 252, 702, 2187, 6642, 19602, 59049, 176904, 532170, 1594323, 4780782, 14351094, 43046721, 129146724, 387400806, 1162261467
Offset: 1

Views

Author

Frank Ruskey and Nate Kube, Aug 19 2002

Keywords

Comments

Same as the number of elements of GF(3^n) with trace 2 and subtrace 2.

Examples

			a(2;1,2)=2. Let GF(3^2) be defined by the field extension GF(3)[x]/( 2+b+b^2 ). The two elements of GF(3^2) with trace 1 and subtrace 2 are { 1+b, 2b }.

Links

Crossrefs

Cf. A074000, A074001, A074002, A074003, A074004.

Programs

  • SageMath
    d = {(0, 0): [1], (0, 1): [0], (0, 2): [0], (1, 0): [1], (1, 1): [0], (1, 2): [0], (2, 0): [1], (2, 1): [0], (2, 2): [0]}
for n in (2..9):
    for a in d.values(): a.append(0)
    k. = GF((3, n))
    for x in k:
        d[(x.trace(), x.charpoly().list()[-3])][-1] += 1
print(d[(1, 2)]) # Andrey Zabolotskiy, Nov 07 2024

Extensions

a(9) and a(14)-a(15) corrected, unverified terms a(17)-a(20) removed. Based on the original Data in A074000-A074005, a(17)-a(20) are possibly equal to 14351094, 43046721, 129146724, 387400806. - Andrey Zabolotskiy, Nov 07 2024
Terms a(17)-a(20) recomputed and added again (verified that all the terms a(17)-a(20) conjectured by Andrey Zabolotskiy are correct), and added term a(21). - Robin Visser, Dec 28 2024

A074001 Number of elements of GF(3^n) with trace 0 and subtrace 1.

Original entry on oeis.org

0, 2, 0, 12, 30, 72, 252, 702, 2268, 6480, 19602, 59292, 176904, 532170, 1592136, 4785156, 14351094, 43040160, 129146724, 387400806, 1162320516
Offset: 1

Views

Author

Frank Ruskey and Nate Kube, Aug 19 2002

Keywords

Examples

			a(2;0,1)=2. Let GF(3^2) be defined by the field extension GF(3)[x]/( 2+b+b^2 ). The two elements of GF(3^2) with trace 0 and subtrace 1 are { 2+b, 1+2b }.

Links

Crossrefs

Cf. A074000, A074002, A074003, A074004, A074005.

Programs

  • Sage
    def a(n):
    ans = 0
    for x in GF(3^n):
        if x.charpoly().coefficients(sparse=False)[-3:-1]==[1, 0]: ans += 1
    return ans  # Robin Visser, Dec 28 2024

Extensions

Terms a(13), a(15), a(16) corrected, unverified terms a(17)-a(20) removed. Based on the original Data in A074000-A074005, a(17)-a(20) are possibly equal to 14351094, 43053282, 129146724, 387400806. - Andrey Zabolotskiy, Nov 11 2024
Terms a(17)-a(20) recomputed and added again (verified that the terms a(17), a(19), a(20) conjectured by Andrey Zabolotskiy are correct), and added term a(21). - Robin Visser, Dec 28 2024

A074002 Number of elements of GF(3^n) with trace 0 and subtrace 2.

Original entry on oeis.org

0, 0, 6, 6, 30, 72, 252, 756, 2106, 6642, 19602, 59292, 176904, 530712, 1596510, 4780782, 14351094, 43040160, 129146724, 387440172, 1162202418
Offset: 1

Views

Author

Frank Ruskey and Nate Kube, Aug 19 2002

Keywords

Links

Crossrefs

Cf. A074000, A074001, A074003, A074004, A074005.

Programs

  • Sage
    def a(n):
    ans = 0
    for x in GF(3^n):
        if x.charpoly().coefficients(sparse=False)[-3:-1]==[2, 0]: ans += 1
    return ans  # Robin Visser, Dec 28 2024

Formula

A074000(n) + A074001(n) + a(n) = 3^(n-1) = A000244(n-1). - R. J. Mathar, Jun 14 2019

Extensions

Formula and terms a(14)-a(15) corrected, unverified terms a(17)-a(20) removed. Based on the original Data in A074000-A074005, a(17)-a(20) are possibly equal to 14351094, 43053282, 129146724, 387440172. - Andrey Zabolotskiy, Nov 08 2024
Terms a(17)-a(20) recomputed and added again (verified that the terms a(17), a(19), a(20) conjectured by Andrey Zabolotskiy are correct), and added term a(21). - Robin Visser, Dec 28 2024

A074003 Number of elements of GF(3^n) with trace 1 and subtrace 0.

Original entry on oeis.org

1, 0, 3, 9, 30, 81, 225, 756, 2187, 6561, 19602, 59049, 177633, 530712, 1594323, 4782969, 14351094, 43046721, 129127041, 387440172, 1162261467
Offset: 1

Views

Author

Frank Ruskey and Nate Kube, Aug 19 2002

Keywords

Comments

Same as the number of elements of GF(3^n) with trace 2 and subtrace 0.

Links

Crossrefs

Cf. A074000, A074001, A074002, A074004, A074005.

Programs

  • Sage
    def a(n):
    if n==1: return 1
    ans = 0
    for x in GF(3^n):
        if x.charpoly().coefficients(sparse=False)[-3:-1]==[0, 1]: ans += 1
    return ans  # Robin Visser, Dec 28 2024

Extensions

Terms a(13)-a(16) corrected, unverified terms a(17)-a(20) removed. Based on the original Data in A074000-A074005, a(17)-a(20) are possibly equal to 14351094, 43046721, 129127041, 387440172. - Andrey Zabolotskiy, Nov 11 2024
Terms a(17)-a(20) recomputed and added again (verified that all the terms a(17)-a(20) conjectured by Andrey Zabolotskiy are correct), and added term a(21). - Robin Visser, Dec 28 2024
Showing 1-5 of 5 results.

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