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 1-8 of 8 results.

A077079 Number of inequivalent bracelets from A006840 with the additional equivalence condition that subsets of 1-beads whose position vectors add to zero can be removed. Different values of vector sums of (-1)^(k/n) with k taking n values in 1..2n up to rotation and reflection.

Original entry on oeis.org

1, 2, 3, 6, 11, 20, 53, 130, 199, 784, 2135, 2649, 15695, 43085, 32764
Offset: 1

Views

Author

Wouter Meeussen, Oct 27 2002

Keywords

Comments

At n=15 the sequence decreases because of the large number of divisors of 30.

Crossrefs

Identical to A077078 up to n=9. Cf. A006840.

Programs

  • Mathematica
    lowest[li_] := First[Sort[Join[NestList[RotateRight, li, 2n-1], NestList[RotateRight, 1-li, 2n-1], NestList[RotateRight, Reverse@li, 2n-1], NestList[RotateRight, 1-Reverse@li, 2n-1]]]]; ker[n_, k_] := Flatten[Table[Join[{1}, 0Range[ -1+2n/k]], {k}]]; ingekort[li_] := Module[{temp, divi}, len=Length[li]; temp=li-(liRotateRight[li, len/2]); divi=First/@FactorInteger[len]; Table[d=divi[[s]]; k=ker[len/2, d]; temp=Fold[kort[ #1, #2]&, temp, NestList[RotateRight, k, len/d-1]], {s, Length[divi], 2, -1}]; lowest[temp]]; kort[q_, k_] := If[(q.k>=Floor[d/2+1])&&(q.RotateRight[k-kq, len/2]===0), q-kq+RotateRight[k-kq, len/2], q]; Length[inequiv=Union[ingekort/@ListOfBraceletsA006840]]

A077078 Number of different absolute values of vector sums generated by n vectors chosen from 2n (equally spaced) points around the unit circle. Dipole moments of n positive and n negative equally spaced charges on a circle.

Original entry on oeis.org

1, 2, 3, 6, 11, 20, 53, 130, 196, 725, 1990, 2437, 14466, 38697, 18878
Offset: 1

Views

Author

Wouter Meeussen, Oct 27 2002

Keywords

Comments

Starting from n ones and n zero's cf. A006840, the simplification can remove subsets of ones with vector sum equal to zero, thus lowering the count of ones in the simplified bracelet.
The count is lower than A077079 because the absolute vector sums can be equal without being symmetrically related.

Examples

			The 35 bracelets for n=6 simplify to the following 20: {0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,1}, {0,0,0,0,0,0,0,0,0,0,1,1}, {0,0,0,0,0,0,0,0,0,1,0,1}, {0,0,0,0,0,0,0,0,0,1,1,1}, {0,0,0,0,0,0,0,0,1,0,0,1}, {0,0,0,0,0,0,0,0,1,0,1,1}, {0,0,0,0,0,0,0,0,1,1,1,1}, {0,0,0,0,0,0,0,1,0,1,0,1}, {0,0,0,0,0,0,0,1,0,1,1,1}, {0,0,0,0,0,0,1,0,0,0,0,1}, {0,0,0,0,0,0,1,0,0,1,0,1}, {0,0,0,0,0,0,1,0,1,0,1,1}, {0,0,0,0,0,0,1,0,1,1,0,1}, {0,0,0,0,0,0,1,1,1,1,1,1}, {0,0,0,0,1,0,0,0,0,1,0,1}, {0,0,0,0,1,0,0,1,0,1,0,1}, {0,0,0,0,1,0,1,0,0,1,0,1}, {0,0,0,0,1,0,1,1,1,1,0,1}, {0,0,1,0,1,0,1,1,0,1,0,1}

Crossrefs

Cf. A077079, A006840.

A045611 Number of different energy states of n positive and n negative charges on a necklace. Different sets of distances between n points chosen from 2n equally spaced points on a circle.

Original entry on oeis.org

1, 1, 2, 3, 7, 13, 35, 85, 254, 701, 2377, 7944, 25220, 95910, 332300, 1164825, 4379920, 16649851, 58473414
Offset: 0

Views

Author

Wouter Meeussen

Keywords

Examples

			a(10) = 2377 because the multiplicities {1; 2; 5; 10; 20; 40; 60; 80; 120} occur {1; 1; 3; 22; 362; 1855; 1; 130; 2} times and 2377 = 1 + 1 + 3 + 22 + 362 + 1855 + 1 + 130 + 2. Total number of configurations = 1*1 + 1*2 + 3*5 + 22*10 + .. = 92378 (see C(2n + 1, n + 1)=A001700).

Links

Crossrefs

For n=0..7 this is equal to A006840. Cf. A045610, A045611, A045612, A006840, A077078, A077079.

Programs

  • Mathematica
    n=4; perm=Permutations[ Table[ -1, {n-1} ]~Join~Table[ 1, {n} ] ]; res=Table[ (Prepend[ #, -1 ].RotateRight[ Prepend[ #, -1 ], k ])/2, {k, 1, n} ]&/@perm; kort=Length[ Length/@Split[ Sort[ res ] ] ]

Extensions

a(15) corrected and a(16)-a(18) from Sean A. Irvine, Mar 17 2021

A045629 Number of 2n-bead black-white complementable necklaces with n black beads.

Original entry on oeis.org

1, 1, 2, 3, 7, 15, 44, 128, 415, 1367, 4654, 16080, 56450, 200170, 716728, 2585850, 9393119, 34319667, 126047906, 465076160, 1723097066, 6407856892, 23910271224, 89493903438, 335912741682, 1264106399934, 4768448177636, 18027218147818
Offset: 0

Views

Author

David W. Wilson

Keywords

Links

Crossrefs

Cf. A006840.

Programs

  • Mathematica
    a[n_] := If[n == 0, 1, (1/(2*n))*DivisorSum[n, EulerPhi[n/#1]*Binomial[2*#1 - 1, #1 - 1] + EulerPhi[2*(n/#1)]*2^(#1 - 1)&]];
Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Oct 08 2017, translated from PARI *)
  • PARI
    a(n) = if(n==0, 1, (1/(2*n)) * sumdiv(n, d, eulerphi(n/d)*binomial(2*d-1, d-1) + eulerphi(2*n/d)*2^(d-1))); \\ Andrew Howroyd, Sep 27 2017

Formula

a(n) = (1/2n) * Sum_{d|n} (phi(n/d)*C(2d-1, d-1) + phi(2n/d)*2^(d-1)). - Christian G. Bower
a(n) ~ 4^(n-1) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Oct 08 2017

A045633 Number of 2n-bead black-white reversible complementable necklaces with n black beads and fundamental period 2n.

Original entry on oeis.org

1, 1, 1, 2, 5, 12, 31, 84, 250, 762, 2504, 8358, 28928, 101339, 361184, 1297864, 4707712, 17179434, 63068079, 232615770, 861723271, 3204236692, 11955827898, 44748176652, 167959114814, 632058070297, 2384234976235, 9013628450510
Offset: 0

Views

Author

David W. Wilson

Keywords

Links

Programs

  • Mathematica
    b[n_] := (1/(2n)) DivisorSum[n, EulerPhi[n/#] Binomial[2# - 1, # - 1] + EulerPhi[2(n/#)] 2^(# - 1) &];
A006840[n_] := If[n == 0, 1, (b[n] + 2^(n - 2) + Binomial[n - Mod[n, 2], Quotient[n, 2]]/2)/2];
a[n_] := If[n == 0, 1, Sum[MoebiusMu[n/d] A006840[d], {d, Divisors[n]}]];
Array[a, 30, 0] (* Jean-François Alcover, Aug 28 2019 *)

Formula

Moebius transform of A006840 (Christian Bower).

A259341 Triangle read by rows: number of balanced twills with 2n harnesses and maximum float length k (1 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 4, 5, 2, 1, 1, 9, 13, 8, 3, 1, 1, 14, 33, 22, 11, 3, 1, 1, 30, 91, 77, 38, 15, 4, 1, 1, 53, 250, 246, 137, 54, 19, 4, 1, 1, 114, 719, 852, 501, 222, 79, 24, 5, 1
Offset: 1

Views

Author

N. J. A. Sloane, Jun 27 2015

Keywords

Examples

			Triangle begins:
  1;
  1,   1;
  1,   1,   1;
  1,   3,   2,   1;
  1,   4,   5,   2,   1;
  1,   9,  13,   8,   3,   1;
  1,  14,  33,  22,  11,   3,  1;
  1,  30,  91,  77,  38,  15,  4,  1;
  1,  53, 250, 246, 137,  54, 19,  4, 1;
  1, 114, 719, 852, 501, 222, 79, 24, 5, 1;
  ...

References

  • J. A. Hoskins, C. E. Praeger and A. P. Street, Balanced twills with bounded float length, Congress. Numerantium, 40 (1983), 77-89.

Links

Crossrefs

Row sums are A006840.

A077013 Number of different argument values of vector sums generated by n vectors chosen from 2n (equally spaced) points around the unit circle. Arguments are considered different only up to rotation and reflection.

Original entry on oeis.org

1, 2, 1, 2, 3, 7, 14, 43, 65, 292, 992, 1154
Offset: 1

Views

Author

Wouter Meeussen, Nov 28 2002

Keywords

Comments

Arg[0] is taken as being equal to 0.

Examples

			The arguments up to rotation and reflection for n=6 are {0, 0.17013, 0.206867, 0.293133, 0.32987, 0.385502, 0.5} or exactly {0, 1/2, 5/2-(6*ArcTan[2])/Pi, 5/2-(6*ArcTan[1+2/Sqrt[3]])/Pi, -3/2-(6*ArcTan[4-3*Sqrt[3]])/Pi, 3/2-(6*ArcTan[6- 3*Sqrt[3]])/Pi, -5/2+(6*ArcTan[4+3*Sqrt[3]])/Pi }

Crossrefs

Cf. A077078, A006840.

A115124 Number of imprimitive (periodic) 2n-bead black-white reversible complementable necklaces with n black beads.

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 4, 1, 7, 3, 14, 1, 40, 1, 86, 15, 257, 1, 797, 1, 2523, 87, 8360, 1, 29218, 13, 101341, 765, 361275, 1, 1300415, 1, 4707969, 8361, 17179436, 97, 63097809, 1, 232615772, 101342, 861726044, 1, 3204597995, 1, 11955836263, 1298641, 44748176654, 1
Offset: 0

Views

Author

Valery A. Liskovets, Jan 17 2006

Keywords

Comments

a(p)=1 for prime p.

Programs

  • Mathematica
    b[n_] := (1/(2n)) DivisorSum[n, EulerPhi[n/#] Binomial[2# - 1, # - 1] + EulerPhi[2(n/#)] 2^(# - 1)&];
A006840[n_] := If[n == 0, 1, (b[n] + 2^(n - 2) + Binomial[n - Mod[n, 2], Quotient[n, 2]]/2)/2];
A045633[n_] := If[n==0, 1, Sum[MoebiusMu[n/d] A006840[d], {d, Divisors[n]}] ];
a[n_] := A006840[n] - A045633[n];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Aug 28 2019 *)

Formula

a(n)=A006840(n) - A045633(n).

Extensions

More terms from Jean-François Alcover, Aug 28 2019
Showing 1-8 of 8 results.

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

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