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.

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A115121 Number of imprimitive (periodic) bracelets (or necklaces) with n red or blue beads such that the beads switch colors when bracelet is turned over.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 4, 2, 5, 1, 11, 1, 11, 5, 22, 1, 37, 1, 64, 11, 95, 1, 210, 4, 317, 30, 625, 1, 1160, 1, 2122, 95, 3857, 13, 7612, 1, 13799, 317, 26518, 1, 50559, 1, 95887, 1124, 182363, 1, 352750, 10, 671150, 3857, 1292764, 1, 2492933, 97, 4797904, 13799
Offset: 1

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Author

Valery A. Liskovets, Jan 17 2006

Keywords

Comments

a(p)=1 for prime p.
Presumably a(n) = A115118(n) = A066656(n)/2 for odd n.

Programs

Formula

a(n) = A053656(n) - A066313(n).

Extensions

More terms from Jean-François Alcover, Aug 28 2019

A284711 Even bisection of A283848.

Original entry on oeis.org

23, 86, 339, 1332, 5298, 21066, 83987, 334966, 1336988, 5338206, 21321234, 85176636, 340338398, 1360073016, 5435820051, 21727481616, 86853790498, 347214198246, 1388133456348, 5549915835836, 22190143855898, 88725807876186, 354775752246802, 1418633882621748, 5672803378074548
Offset: 2

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Author

N. J. A. Sloane, Apr 01 2017

Keywords

Links

Crossrefs

The 8 sequences in Table 8 of Fujita (2017) are A053656, A000011, A256216, A256217, A123045, A283846, A283847, A283848.
The even bisection of A283848 gives A000079.

Programs

  • Maple
    f:= proc(n) uses numtheory;
(4*n)^(-1)*add(phi(d)*4^(2*n/d),d=select(type,divisors(2*n),even))+5*2^(2*n-2)
end proc:
map(f, [$2..40]);
  • PARI
    A(m,n) = if (m%2, 2^((m-1)/2)*n^((m+1)/2), sumdiv(m, d, ((d%2)==0)*(eulerphi(d)*2^(m/d)*n^(m/d)))/(2*m) + 2^(m/2-2)*n^(m/2)*(2*n+1));
lista(nn) = for(n=2, nn, print1(T(2*n, 2), ", ")) \\ Michel Marcus, Apr 02 2017

Formula

a(n) = A283848(2*n)=(4*n)^(-1)*Sum_{d|2*n, d even} phi(d)*4^(2*n/d) + 5*2^(2*n-2). - Robert Israel, Aug 23 2018 after Fujita (2017), Eq. (101)

Extensions

More terms from Michel Marcus, Apr 02 2017
Edited by Robert Israel, Aug 23 2018

A058880 a(n) = (1/(2n)) * Sum_{d|n} phi(d) * 2^(2n/d) + (2^((n-4)/2), if n is even).

Original entry on oeis.org

12, 36, 104, 352, 1172, 4119, 14572, 52492, 190652, 699266, 2581112, 9587602, 35791472, 134219859, 505290272, 1908881998, 7233629132, 27487817244, 104715393912, 399822505942, 1529755308212, 5864062368274, 22517998136936
Offset: 3

Views

Author

N. J. A. Sloane, Jan 07 2001

Keywords

Comments

Previous name was "Number of orientations of an n-cycle". Apparently, the book by Harary and Palmer erroneously gives this formula for the number of orientations of an n-cycle, but the correct sequence for that is A053656. The error is in the exponent of 2 in the sum; it should be n/d, not 2*n/d. - Pontus von Brömssen, Mar 30 2022

References

  • F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 129, (5.3.3).

Links

Crossrefs

Cf. A053656.

Programs

  • Maple
    A058880 := proc(n) local d,t1,t2; if n mod 2 = 0 then t1 := 2^((n-4)/2) else t1 := 0; fi; t2 := divisors(n); for d in t2 do t1 := t1+phi(d)*2^(2*n/d)/(2*n); od; t1; end;
  • Mathematica
    a[n_] := DivisorSum[n, EulerPhi[#]*2^(2n/#) &]/(2n) + If[OddQ[n], 0, 2^((n - 4)/2)]; Array[a, 23, 3] (* Amiram Eldar, Aug 28 2019 *)
  • PARI
    a(n) = sumdiv(n, d, eulerphi(d)*2^(2*n/d))/(2*n) + if (!(n%2), 2^((n-4)/2)); \\ Michel Marcus, Aug 29 2019

Formula

a(n) = (1/(2n)) * Sum_{d|n} phi(d) * 2^(2n/d) + (2^((n-4)/2), if n is even). - Amiram Eldar, Aug 28 2019

Extensions

New name, using existing formula, from Pontus von Brömssen, Mar 30 2022
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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