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 1-6 of 6 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

A073950 Number of strings over Z_3 of length n with trace 1 and subtrace 0.

Original entry on oeis.org

1, 2, 3, 9, 30, 81, 225, 702, 2187, 6561, 19602, 59049, 177633, 532170, 1594323, 4782969, 14351094, 43046721, 129127041, 387400806, 1162261467, 3486784401, 10460294154, 31381059609, 94143533121, 282430067922, 847288609443, 2541865828329, 7625599079310
Offset: 1

Views

Author

Frank Ruskey and Nate Kube, Aug 15 2002

Keywords

Comments

Same as number of strings over Z_3 of length n with trace 2 and subtrace 0. Same as number of strings over GF(3) of length n with trace 1 and subtrace 0. Same as number of strings over GF(3) of length n with trace 2 and subtrace 0.

Links

Crossrefs

Cf. A073947, A073948, A073949, A073951, A073952.

Programs

  • Mathematica
    LinearRecurrence[{6, -15, 27, -36, 27}, {1, 2, 3, 9, 30}, 30] (* Jean-François Alcover, Jan 07 2019 *)

Formula

a(n; t, s) = a(n-1; t, s) + a(n-1; t+2, s+2t+1) + a(n-1; t+1, s+t+1) where t is the trace and s is the subtrace.
G.f.: q*(q-1)*(3*q^3-3*q^2+3*q-1)/[(1-3q)(1+3q^2)(1-3q+3q^2)]. - Lawrence Sze, Oct 24 2004

Extensions

Terms a(21) onward from Max Alekseyev, Apr 09 2013

A073948 Number of strings over Z_3 of length n with trace 0 and subtrace 1.

Original entry on oeis.org

0, 0, 0, 6, 30, 90, 252, 756, 2268, 6642, 19602, 58806, 176904, 530712, 1592136, 4780782, 14351094, 43053282, 129146724, 387440172, 1162320516, 3486843450, 10460294154, 31380882462, 94143001680, 282429005040, 847287015120, 2541864234006, 7625599079310
Offset: 1

Views

Author

Frank Ruskey and Nate Kube, Aug 15 2002

Keywords

Comments

Same as number of strings over GF(3) of length n with trace 0 and subtrace 1.

Links

Crossrefs

Cf. A073947, A073949, A073950, A073951, A073952.

Formula

a(n; t, s) = a(n-1; t, s) + a(n-1; t+2, s+2t+1) + a(n-1; t+1, s+t+1) where t is the trace and s is the subtrace.
G.f.: -6q^4(q-1)/[(1-3q)(1+3q^2)(1-3q+3q^2)]. - Lawrence Sze, Oct 24 2004

Extensions

Terms a(21) onward from Max Alekseyev, Apr 09 2013

A073949 Number of strings over Z_3 of length n with trace 0 and subtrace 2.

Original entry on oeis.org

0, 2, 6, 12, 30, 90, 252, 702, 2106, 6480, 19602, 58806, 176904, 532170, 1596510, 4785156, 14351094, 43053282, 129146724, 387400806, 1162202418, 3486725352, 10460294154, 31380882462, 94143001680, 282430067922, 847290203766, 2541867422652, 7625599079310
Offset: 1

Views

Author

Frank Ruskey and Nate Kube, Aug 15 2002

Keywords

Comments

Same as number of strings over GF(3) of length n with trace 0 and subtrace 2.

Links

Crossrefs

Cf. A073947, A073948, A073950, A073951, A073952.

Formula

a(n; t, s) = a(n-1; t, s) + a(n-1; t+2, s+2t+1) + a(n-1; t+1, s+t+1) where t is the trace and s is the subtrace.
G.f.: -2q^2(3*q^3-3*q^2+3*q-1)/[(1-3q)(1+3q^2)(1-3q+3q^2)]. - Lawrence Sze, Oct 24 2004

Extensions

Terms a(21) onward from Max Alekseyev, Apr 09 2013

A073951 Number of strings over Z_3 of length n with trace 1 and subtrace 1.

Original entry on oeis.org

0, 1, 3, 6, 21, 81, 252, 729, 2187, 6642, 19845, 59049, 176904, 531441, 1594323, 4780782, 14344533, 43046721, 129146724, 387420489, 1162261467, 3486843450, 10460471301, 31381059609, 94143001680, 282429536481, 847288609443, 2541864234006, 7625594296341
Offset: 1

Views

Author

Frank Ruskey and Nate Kube, Aug 15 2002

Keywords

Comments

Same as number of strings over Z_3 of length n with trace 2 and subtrace 1. Same as number of strings over GF(3) of length n with trace 1 and subtrace 1. Same as number of strings over GF(3) of length n with trace 2 and subtrace 1.

Examples

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

Links

Crossrefs

Cf. A073947, A073948, A073949, A073950, A073952.

Formula

a(n; t, s) = a(n-1; t, s) + a(n-1; t+2, s+2t+1) + a(n-1; t+1, s+t+1) where t is the trace and s is the subtrace.
G.f.: q^2(3*q^3+3*q^2-3*q+1)/[(1-3q)(1+3q^2)(1-3q+3q^2)]. - Lawrence Sze, Oct 24 2004

Extensions

Terms a(21) onward from Max Alekseyev, Apr 09 2013

A073952 Number of strings over Z_3 of length n with trace 1 and subtrace 2.

Original entry on oeis.org

0, 0, 3, 12, 30, 81, 252, 756, 2187, 6480, 19602, 59049, 176904, 530712, 1594323, 4785156, 14351094, 43046721, 129146724, 387440172, 1162261467, 3486725352, 10460294154, 31381059609, 94143001680, 282429005040, 847288609443, 2541867422652, 7625599079310
Offset: 1

Views

Author

Frank Ruskey and Nate Kube, Aug 15 2002

Keywords

Comments

Same as number of strings over Z_3 of length n with trace 2 and subtrace 2. Same as number of strings over GF(3) of length n with trace 1 and subtrace 2. Same as number of strings over GF(3) of length n with trace 2 and subtrace 2.

Examples

			a(3;1,2)=3 since the three ternary strings of trace 1, subtrace 2 and length 3 are { 112, 121, 211 }.

Links

Crossrefs

Cf. A073947, A073948, A073949, A073950, A073951.

Programs

  • Mathematica
    LinearRecurrence[{6,-15,27,-36,27},{0,0,3,12,30},30] (* Harvey P. Dale, Oct 22 2019 *)

Formula

a(n; t, s) = a(n-1; t, s) + a(n-1; t+2, s+2t+1) + a(n-1; t+1, s+t+1) where t is the trace and s is the subtrace.
G.f.: 3q^3(q^2-2*q+1)/[(1-3q)(1+3q^2)(1-3q+3q^2)]. - Lawrence Sze, Oct 24 2004

Extensions

Terms a(21) onward from Max Alekseyev, Apr 09 2013
Showing 1-6 of 6 results.

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