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 1-9 of 9 results.

A073995 Number of strings of length n over GF(4) with trace 0 and subtrace 0.

Original entry on oeis.org

1, 1, 7, 28, 76, 256, 1072, 4288, 16576, 65536, 262912, 1051648, 4197376, 16777216, 67121152, 268484608, 1073790976, 4294967296, 17180065792, 68720263168, 274878693376, 1099511627776, 4398049656832, 17592198627328, 70368756760576, 281474976710656, 1125899957174272, 4503599828697088, 18014398710808576
Offset: 1

Views

Author

Frank Ruskey and Nate Kube, Aug 16 2002

Keywords

Links

Crossrefs

Cf. A073996, A073997, A073998, A073999, A074000.

Formula

a(n; t, s) = a(n-1; t, s) + a(n-1; t-1, s-(t-1)) + a(n-1; t-2, s-2(t-2)) + a(n-1; t-3, s-3(t-3)) where t is the trace and s is the subtrace. Note that all operations involving operands t or s are carried out over GF(4).
G.f.: -q*(26*q^3+5*q-1-13*q^2)/[(1-2q)(1-4q)(1+4q^2)]. - Lawrence Sze, Oct 24 2004

Extensions

More terms from Max Alekseyev, Apr 16 2013

A073996 Number of strings of length n over GF(4) with trace 0 and subtrace 1.

Original entry on oeis.org

0, 1, 3, 12, 60, 256, 1008, 4032, 16320, 65536, 261888, 1047552, 4193280, 16777216, 67104768, 268419072, 1073725440, 4294967296, 17179803648, 68719214592, 274877644800, 1099511627776, 4398045462528, 17592181850112, 70368739983360, 281474976710656, 1125899890065408, 4503599560261632, 18014398442373120
Offset: 1

Views

Author

Frank Ruskey and Nate Kube, Aug 16 2002

Keywords

Comments

Same as the number of strings of length n over GF(4) with trace 0 and subtrace x where x=RootOf(z^2+z+1). Same as the number of strings of length n over GF(4) with trace 0 and subtrace y where y=1+x.

Examples

			a(2;0,1)=1 since the one 4-ary string of trace 0, subtrace 1 and length 2 is { 11 }.

Links

Crossrefs

Cf. A073995, A073997, A073998, A073999, A074000.

Programs

  • Mathematica
    CoefficientList[Series[x^2 (6x^2-3x+1)/((1-2x)(1-4x)(1+4x^2)), {x,0,30}], x]  (* Harvey P. Dale, Apr 03 2011 *)

Formula

a(n; t, s) = a(n-1; t, s) + a(n-1; t-1, s-(t-1)) + a(n-1; t-2, s-2(t-2)) + a(n-1; t-3, s-3(t-3)) where t is the trace and s is the subtrace. Note that all operations involving operands t or s are carried out over GF(4).
G.f.: (6*q^2-3*q+1)*q^2/[(1-2q)(1-4q)(1+4q^2)]. - Lawrence Sze, Oct 24 2004

Extensions

More terms from Harvey P. Dale, Apr 03 2011
More terms from Max Alekseyev, Apr 16 2013

A073997 Number of strings of length n over GF(4) with trace 1 and subtrace 0.

Original entry on oeis.org

1, 2, 3, 16, 76, 272, 1008, 4096, 16576, 65792, 261888, 1048576, 4197376, 16781312, 67104768, 268435456, 1073790976, 4295032832, 17179803648, 68719476736, 274878693376, 1099512676352, 4398045462528, 17592186044416, 70368756760576, 281474993487872, 1125899890065408, 4503599627370496, 18014398710808576
Offset: 1

Views

Author

Frank Ruskey and Nate Kube, Aug 16 2002

Keywords

Comments

Same as the number of strings of length n over GF(4) with trace x and subtrace 0 where x=RootOf(z^2+z+1). Same as the number of strings of length n over GF(4) with trace y and subtrace 0 where y=1+x.

Links

Crossrefs

Cf. A073995, A073996, A073998, A073999, A074000.

Formula

a(n; t, s) = a(n-1; t, s) + a(n-1; t-1, s-(t-1)) + a(n-1; t-2, s-2(t-2)) + a(n-1; t-3, s-3(t-3)) where t is the trace and s is the subtrace. Note that all operations involving operands t or s are carried out over GF(4).
G.f.: -(2*q^3-3*q^2+4*q-1)*q/[(1-2q)(1-4q)(1+4q^2)]. - Lawrence Sze, Oct 24 2004

Extensions

More terms from Max Alekseyev, Apr 16 2013

A073998 Number of strings of length n over GF(4) with trace 1 and subtrace 1.

Original entry on oeis.org

0, 2, 7, 16, 60, 272, 1072, 4096, 16320, 65792, 262912, 1048576, 4193280, 16781312, 67121152, 268435456, 1073725440, 4295032832, 17180065792, 68719476736, 274877644800, 1099512676352, 4398049656832, 17592186044416, 70368739983360, 281474993487872, 1125899957174272, 4503599627370496, 18014398442373120
Offset: 1

Views

Author

Frank Ruskey and Nate Kube, Aug 16 2002

Keywords

Comments

Same as the number of strings of length n over GF(4) with trace x and subtrace y where x=RootOf(z^2+z+1) and y=1+x. Same as the number of strings of length n over GF(4) with trace y and subtrace x.

Examples

			a(2; x,y)=2 since the two 4-ary strings of trace x, subtrace y and length 2 are { 1y, y1 }.

Links

Crossrefs

Cf. A073995, A073996, A073997, A073999, A074000.

Formula

a(n; t, s) = a(n-1; t, s) + a(n-1; t-1, s-(t-1)) + a(n-1; t-2, s-2(t-2)) + a(n-1; t-3, s-3(t-3)) where t is the trace and s is the subtrace. Note that all operations involving operands t or s are carried out over GF(4).
G.f.: -(2*q^2+5*q-2)*q^2/[(1-2q)(1-4q)(1+4q^2)]. - Lawrence Sze, Oct 24 2004

Extensions

More terms from Max Alekseyev, Apr 16 2013

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

A074004 Number of elements of GF(3^n) with trace 1 and subtrace 1.

Original entry on oeis.org

0, 1, 3, 12, 21, 81, 252, 729, 2187, 6480, 19845, 59049, 176904, 531441, 1594323, 4785156, 14344533, 43046721, 129146724, 387420489, 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 1.

Examples

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

Links

Crossrefs

Cf. A074000, A074001, A074002, A074003, A074005.

Programs

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

Extensions

a(14) corrected, unverified terms a(17)-a(20) removed. Based on the original Data in A074000-A074005, a(17)-a(20) are possibly equal to 14344533, 43046721, 129146724, 387420489. - 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 1-9 of 9 results.

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

Content is available under The OEIS End-User License Agreement.