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.

User: Zoran Sunic

Zoran Sunic's wiki page.

Zoran Sunic has authored 5 sequences.

A294781 Growth of the Lamplighter group: number of elements in the Lamplighter group Z wr Z of length up to n with respect to the standard generating set {a,t}.

Original entry on oeis.org

1, 5, 17, 53, 153, 421, 1125, 2937, 7537, 19093, 47881, 119133, 294585, 724869, 1776717, 4341425, 10582177, 25743269, 62527553, 151682821, 367594457, 890137893, 2154129717, 5210373929, 12597758737, 30449544885, 73580024633, 177767884973, 429416696185, 1037172672005, 2504846014621
Offset: 0

Views

Author

Zoran Sunic, Nov 08 2017

Keywords

Comments

The group is presented by .

Examples

			a(2)=17, since the elements of length up to 2 are 1, a, a^-1, t, t^-1, a^2, at, at^-1, a^-2, a^-1t, a^-1t^-1, ta, ta^-1, t^2, t^-1a, t^-1a^-1, t^-2.

Links

Crossrefs

Cf. A294683. Partial sums of A294782.

Programs

  • Mathematica
    CoefficientList[ Series[-((x^2 + 1) (x - 1)^2 (x + 1)^3)/((x^3 + x^2 + x - 1)^2 (x^2 + 2 x - 1)), {x, 0, 27}], x] (* or *)
LinearRecurrence[{4, -2, -4, -4, 4, 6, 4, 1}, {1, 5, 17, 53, 153, 421, 1125, 2937}, 28] (* Robert G. Wilson v, Aug 08 2018 *)

Formula

G.f.: (1-x)^2 (1+x)^3 (1+x^2) / ((1-2x-x^2)(1-x-x^2-x^3)^2).

A294782 Spherical growth of the Lamplighter group: number of elements in the Lamplighter group Z wr Z of length n with respect to the standard generating set {a,t}.

Original entry on oeis.org

1, 4, 12, 36, 100, 268, 704, 1812, 4600, 11556, 28788, 71252, 175452, 430284, 1051848, 2564708, 6240752, 15161092, 36784284, 89155268, 215911636, 522543436, 1263991824, 3056244212, 7387384808, 17851786148, 43130479748, 104187860340, 251648811212, 607755975820, 1467673342616
Offset: 0

Views

Author

Zoran Sunic, Nov 08 2017

Keywords

Comments

The group is presented by .

Examples

			a(2)=12, since the elements of length 2 are a^2, at, at^-1, a^-2, a^-1t, a^-1t^-1, ta, ta^-1, t^2, t^-1a, t^-1a^-1, t^-2.

Links

Crossrefs

Cf. A288348. First differences of A294781.

Programs

  • Mathematica
    LinearRecurrence[{4,-2,-4,-4,4,6,4,1},{1,4,12,36,100,268,704,1812,4600},40] (* Harvey P. Dale, Jan 31 2025 *)

Formula

G.f.: (1-x)^3 (1+x)^3 (1+x^2) / ((1-2x-x^2)(1-x-x^2-x^3)^2).

A294683 Growth of the Lamplighter group: number of elements in the Lamplighter group L_2 = Z/2Z wr Z of length up to n with respect to the standard generating set {a,t}.

Original entry on oeis.org

1, 4, 10, 22, 44, 84, 155, 278, 490, 850, 1457, 2474, 4167, 6974, 11609, 19238, 31762, 52274, 85806, 140534, 229735, 374958, 611158, 995016, 1618409, 2630222, 4271663, 6933430, 11248251, 18240668, 29569464, 47920016, 77639264, 125763290, 203680213, 329821130, 534014584
Offset: 0

Views

Author

Zoran Sunic, Nov 06 2017

Keywords

Comments

The group is presented by L_2 = .

Examples

			a(2)=10, since the elements of length up to 2 are 1, a, t, t^-1, at, at^-1, ta, t^2, t^-1a, t^-2.

Links

Crossrefs

Partial sums of A288348.

Programs

  • Mathematica
    CoefficientList[ Series[((x^2 + x + 1) (x - 1) (x + 1)^3)/((x^3 + x^2 - 1)^2 (x^2 + x - 1)), {x, 0, 36}], x] (* or *)
LinearRecurrence[{1, 3, 0, -5, -3, 2, 3, 1}, {1, 4, 10, 22, 44, 84, 155, 278}, 37] (* Robert G. Wilson v, Aug 08 2018 *)
  • PARI
    Vec((1-x)*(1+x)^3*(1+x+x^2)/((1-x-x^2)*(1-x^2-x^3)^2) + O(x^40)) \\ Michel Marcus, Nov 07 2017

Formula

G.f.: (1-x)(1+x)^3(1+x+x^2) / ((1-x-x^2)(1-x^2-x^3)^2).

Extensions

More terms from Michel Marcus, Nov 07 2017

A293958 Smallest odd prime divisor of (2n+1)^2 + 1.

Original entry on oeis.org

5, 13, 5, 41, 61, 5, 113, 5, 181, 13, 5, 313, 5, 421, 13, 5, 613, 5, 761, 29, 5, 1013, 5, 1201, 1301, 5, 17, 5, 1741, 1861, 5, 2113, 5, 2381, 2521, 5, 29, 5, 3121, 17, 5, 3613, 5, 17, 41, 5, 4513, 5, 13, 5101, 5, 37, 5, 13, 61, 5, 17, 5, 73, 7321, 5, 13, 5, 53, 8581, 5, 13, 5, 9661, 9941, 5
Offset: 1

Views

Author

N. J. A. Sloane, Nov 04 2017, following a suggestion from Zoran Sunic

Keywords

Comments

If the map "x -> smallest odd prime divisor of n^2+1" is iterated, does it always terminate in the 2-cycle (5 <-> 13)? - Zoran Sunic, Oct 25 2017
A027862 is a subsequence. - David A. Corneth, Nov 04 2017

Links

Crossrefs

A bisection of A125256. Cf. A027862, A069894, A078701, A256970.

Programs

  • Mathematica
    sod[n_]:=With[{fi=FactorInteger[n]},If[fi[[1,1]]==2,fi[[2,1]],fi[1,1]]]; sod/@(Range[3,151,2]^2+1) (* Harvey P. Dale, Dec 23 2023 *)
  • PARI
    a(n) = factor((2*n+1)^2 + 1)[2,1]; \\ Michel Marcus, Nov 04 2017

Formula

a(n) = A078701(A069894(n)). - Michel Marcus, Nov 04 2017

A071962 Number of double points of the map that, for each term t of a sequence, counts the preceding terms that are greater than or equal to t.

Original entry on oeis.org

1, 2, 4, 10, 26, 70, 216, 682, 2264, 7960, 29262, 113256, 452586, 1886306, 8109828, 36274448, 167157176
Offset: 0

Views

Author

Zoran Sunic, Jun 24 2002

Keywords

Comments

It would be nice to have a formula or recurrence!

Examples

			The ten double points for n=3 form the following 5 pairs: (0000,0123), (0003,0120), (0020,0103), (0023,0100), (0021,0101)

Links

Extensions

More terms from John W. Layman, Jul 01 2002

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

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