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.

A054660 Number of monic irreducible polynomials over GF(4) of degree n with fixed nonzero trace.

Original entry on oeis.org

1, 2, 5, 16, 51, 170, 585, 2048, 7280, 26214, 95325, 349520, 1290555, 4793490, 17895679, 67108864, 252645135, 954437120, 3616814565, 13743895344, 52357696365, 199911205050, 764877654105, 2932031006720, 11258999068416
Offset: 1

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Author

N. J. A. Sloane, Apr 18 2000

Keywords

Comments

Also number of Lyndon words of length n with trace 1 over GF(4).
Let x = RootOf( z^2+z+1 ) and y = 1+x. Also number of Lyndon words of length n with trace x over GF(4). Also number of Lyndon words of length n with trace y over GF(4).
Also number of 4-ary Lyndon words (i.e., Lyndon words over Z_4) of length n with trace 1 (mod 4). Also the same with trace 3 (mod 4). - Andrey Zabolotskiy, Dec 19 2020

Examples

			a(3; y)=5 since the five Lyndon words over GF(4) of trace y and length 3 are { 00y, 01x, 0x1, 11y, xxy }; the five Lyndon words over Z_4 of trace 1 (mod 4) and length 3 are { 001, 023, 032, 113, 122 }.

Links

Crossrefs

Cf. A000048, A051841, A046211, A046209, A054661, etc.
Cf. A008683, A054661, A027377, A300628, A074033, A074034, A074035, A074448, A074449, A074450.
Cf. A074406, A074407, A074408, A074409.

Formula

From Seiichi Manyama, Mar 11 2018: (Start)
a(n) = A000048(2*n) = (1/(4*n)) * Sum_{odd d divides n} mu(d)*4^(n/d), where mu is the Möbius function A008683.
a(n+1) = A300628(n,n) for n >= 0. (End)
From Andrey Zabolotskiy, Dec 19 2020: (Start)
a(n) = A074033(n) + A074034(n) + 2 * A074035(n).
a(n) = A074448(n) + A074449(n) + 2 * A074450(n).
a(n) = A074406(n) + A074407(n) + A074408(n) + A074409(n). (End)

Extensions

More terms from James Sellers, Apr 19 2000

A074450 Let x = RootOf(z^2 + z + 1) and y = 1+x. Number of 4-ary Lyndon words of length n over GF(4) with trace 1 and subtrace x.

Original entry on oeis.org

0, 0, 1, 4, 12, 40, 144, 512, 1813, 6528, 23808, 87380, 322560, 1198080, 4473647, 16777216, 63160320, 238605640, 904200192, 3435973836, 13089411609, 49977753600, 191219367936, 733007751680, 2814749599332, 10825959997440, 41699995927744, 160842843834660, 621186153185280
Offset: 1

Views

Author

Frank Ruskey and Nate Kube, Aug 23 2002

Keywords

Comments

Also the number of 4-ary Lyndon words of length n over GF(4) with trace 1 and subtrace y. Also the number of 4-ary Lyndon words of length n over GF(4) with trace x and subtrace 1. Also the number of 4-ary Lyndon words of length n over GF(4) with trace x and subtrace x. Also the number of 4-ary Lyndon words of length n over GF(4) with trace y and subtrace 1. Also the number of 4-ary Lyndon words of length n over GF(4) with trace y and subtrace y.
Is this a duplicate of A074032? - R. J. Mathar, Dec 15 2020

Links

Crossrefs

Cf. A074446, A074447, A074448, A074449.
Cf. A054660, A073995, A073996, A073997, A073998, A073999.

Extensions

Terms a(16) and beyond from Andrey Zabolotskiy, Jul 21 2021

A074446 Number of 4-ary Lyndon words of length n over GF(4) with trace 0 and subtrace 0.

Original entry on oeis.org

1, 0, 2, 6, 15, 40, 153, 528, 1841, 6528, 23901, 87550, 322875, 1198080, 4474738, 16779264, 63164175, 238605640, 904213989, 3436000050, 13089461538, 49977753600, 191219550297, 733008101200, 2814750270420, 10825959997440, 41699998413248, 160842848628150, 621186162441675
Offset: 1

Views

Author

Frank Ruskey and Nate Kube, Aug 23 2002

Keywords

Examples

			a(3;0,0)=2 since the two 4-ary Lyndon words of trace 0, subtrace 0 and length 3 are { 123, 132 }.

Links

Crossrefs

Cf. A074447, A074448, A074449, A074450.
Cf. A054661, A073995, A073996, A073997, A073998, A073999.

Extensions

Terms a(16) and beyond from Andrey Zabolotskiy, Jul 21 2021

A074447 Number of 4-ary Lyndon words of length n over GF(4) with trace 0 and subtrace 1.

Original entry on oeis.org

0, 0, 1, 2, 12, 40, 144, 496, 1813, 6528, 23808, 87210, 322560, 1198080, 4473647, 16775168, 63160320, 238605640, 904200192, 3435947622, 13089411609, 49977753600, 191219367936, 733007402160, 2814749599332, 10825959997440, 41699995927744, 160842839041170, 621186153185280
Offset: 1

Views

Author

Frank Ruskey and Nate Kube, Aug 23 2002

Keywords

Comments

Let x = RootOf( z^2+z+1 ) and y = 1+x. Also the number of 4-ary Lyndon words of length n over GF(4) with trace 0 and subtrace x. Also the number of 4-ary Lyndon words of length n over GF(4) with trace 0 and subtrace y.

Examples

			a(4;0,1)=2 since the two 4-ary Lyndon words of trace 0, subtrace 1 and length 4 are { 0011, 11xx }, where x = RootOf( z^2+z+1 ).

Links

Crossrefs

Cf. A074446, A074448, A074449, A074450.
Cf. A054661, A073995, A073996, A073997, A073998, A073999.

Extensions

Terms a(16) and beyond from Andrey Zabolotskiy, Jul 21 2021

A074449 Number of 4-ary Lyndon words of length n over GF(4) with trace 1 and subtrace 1.

Original entry on oeis.org

0, 1, 2, 4, 12, 45, 153, 512, 1813, 6579, 23901, 87380, 322560, 1198665, 4474738, 16777216, 63160320, 238612920, 904213989, 3435973836, 13089411609, 49977848925, 191219550297, 733007751680, 2814749599332, 10825961287995, 41699998413248, 160842843834660, 621186153185280
Offset: 1

Views

Author

Frank Ruskey and Nate Kube, Aug 23 2002

Keywords

Comments

Let x = RootOf( z^2+z+1 ) and y = 1+x. Also the number of 4-ary Lyndon words of length n over GF(4) with trace x and subtrace y. Also the number of 4-ary Lyndon words of length n over GF(4) with trace y and subtrace x.

Examples

			Let x = RootOf( z^2+z+1 ) and y = 1+x. a(2; y,x)=1 since the one 4-ary Lyndon word of trace y, subtrace x and length 2 is { 1x }.

Links

Crossrefs

Cf. A074446, A074447, A074448, A074450.
Cf. A054660, A073995, A073996, A073997, A073998, A073999.

Extensions

Terms a(16) and beyond from Andrey Zabolotskiy, Jul 21 2021
Showing 1-5 of 5 results.

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

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