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

A073999 Number of strings of length n over GF(4) with trace 1 and subtrace x where x = RootOf(z^2+z+1).

Original entry on oeis.org

0, 0, 3, 16, 60, 240, 1008, 4096, 16320, 65280, 261888, 1048576, 4193280, 16773120, 67104768, 268435456, 1073725440, 4294901760, 17179803648, 68719476736, 274877644800, 1099510579200, 4398045462528, 17592186044416, 70368739983360, 281474959933440, 1125899890065408, 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 1. Same as the number of strings of length n over GF(4) with trace y and subtrace 1 where y = 1+x. Same as the number of strings of length n over GF(4) with trace 1 and subtrace y. Same as the number of strings of length n over GF(4) with trace x and subtrace x. Same as the number of strings of length n over GF(4) with trace y and subtrace y.

Links

Crossrefs

Cf. A073995, A073996, A073997, A073998.

Programs

  • Mathematica
    LinearRecurrence[{6,-12,24,-32},{0,0,3,16},30] (* Harvey P. Dale, Mar 12 2019 *)

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)*q^3/[(1-2q)(1-4q)(1+4q^2)]. - Lawrence Sze, Oct 24 2004
a(n) = -2^(n-3) +( (-2i)^n + (2i)^n +4^n )/16 with i=sqrt(-1). - R. J. Mathar, Nov 18 2011

Extensions

More terms from Max Alekseyev, Apr 16 2013

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

A074448 Number of 4-ary Lyndon words of length n over GF(4) with trace 1 and subtrace 0.

Original entry on oeis.org

1, 1, 1, 4, 15, 45, 144, 512, 1841, 6579, 23808, 87380, 322875, 1198665, 4473647, 16777216, 63164175, 238612920, 904200192, 3435973836, 13089461538, 49977848925, 191219367936, 733007751680, 2814750270420, 10825961287995, 41699995927744, 160842843834660, 621186162441675
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 0. Also the number of 4-ary Lyndon words of length n over GF(4) with trace y and subtrace 0.

Links

Crossrefs

Cf. A074446, A074447, A074449, A074450.
Cf. A054660, 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 1-9 of 9 results.

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

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