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.

A003437 Number of unlabeled Hamiltonian circuits on n-octahedron (cross polytope); also number of circular chord diagrams with n chords, modulo symmetries.

Original entry on oeis.org

0, 1, 2, 7, 29, 196, 1788, 21994, 326115, 5578431, 107026037, 2269254616, 52638064494, 1325663757897, 36021577975918, 1050443713185782, 32723148860301935, 1084545122297249077, 38105823782987999742, 1414806404051118314077
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

See A003436 for labeled case.
See also A278990, A007474.

Programs

  • Mathematica
    nn = 20; M = Array[0&, {2nn, 2nn}];
Mget[n_, k_] := Which[n < 0, 0, n==0, 1, n==1, 1-Mod[k, 2], n==2, k - Mod[k, 2], True, M[[n, k]]];
Mset[n_, k_, v_] := (M[[n, k]] = v);
Minit = Module[{tmp = 0}, For[n = 3, n <= 2nn, n++, For[k = 1, k <= 2nn, k++, tmp = If[OddQ[k], k(n-1) Mget[n-2, k] + Mget[n-4, k], Mget[n-1, k] + k(n-1) Mget[n-2, k] - Mget[n-3, k] + Mget[n-4, k]]; Mset[n, k, tmp]]]];
A007474[n_] := Sum[EulerPhi[d] (Mget[2n/d, d] - Mget[2n/d - 2, d]), {d, Divisors[2n]}]/(2n);
a[n_] := A007474[n]/2 + (Mget[n, 2] - Mget[n-1, 2] + Mget[n-2, 2])/4;
Array[a, nn] (* Jean-François Alcover, Aug 12 2018, after Gheorghe Coserea *)
  • PARI
    N = 20; M = matrix(2*N, 2*N);
Mget(n,k) = { if (n<0, 0, n==0, 1, n==1, 1-(k%2), n==2, k-(k%2), M[n,k]) };
Mset(n,k,v) = { M[n,k] = v;};
Minit() = {
  my(tmp = 0);
  for (n=3, 2*N, for(k=1, 2*N,
    tmp = if (k%2, k*(n-1) * Mget(n-2, k) + Mget(n-4, k),
    Mget(n-1, k) + k*(n-1) * Mget(n-2, k) - Mget(n-3, k) + Mget(n-4, k));
    Mset(n, k, tmp)));
};
Minit();
A007474(n) = sumdiv(2*n, d, eulerphi(d) * (Mget(2*n/d, d) - Mget(2*n/d-2, d)))/(2*n);
a(n) = A007474(n)/2 + (Mget(n,2) - Mget(n-1,2) + Mget(n-2,2))/4;
vector(N, n, a(n))  \\ Gheorghe Coserea, Dec 10 2016

Formula

a(n) ~ 2^(n-3/2) * n^(n-1) / exp(n+1). - Vaclav Kotesovec, Dec 10 2016

A348816 a(n) = number of loopless diagrams by number of K_4 up to rotational symmetry.

Original entry on oeis.org

0, 1, 15, 4790, 4151415, 6619291247, 17510518983528, 71631394311300461, 429426878302882412435, 3616596939726424941979785
Offset: 1

Views

Author

Michael De Vlieger, Nov 01 2021

Keywords

Links

Crossrefs

Cf. A190830, A348815, A348817; A007474, A348814, A348819, A348822.

A348819 a(n) = number of loopless diagrams by number of K_5 up to rotational symmetry.

Original entry on oeis.org

0, 1, 60, 222477, 2211192688, 48357603758012, 2059392303708166507, 155876203880714141444480
Offset: 1

Views

Author

Michael De Vlieger, Nov 01 2021

Keywords

Links

Crossrefs

Cf. A190833, A348818, A348820; A007474, A348814, A348816, A348822.

A348822 a(n) = number of loopless diagrams by number of K_6 up to rotational symmetry.

Original entry on oeis.org

0, 1, 335, 11508322, 1324603148183, 404760320241653655, 282780723811372935744420
Offset: 1

Views

Author

Michael De Vlieger, Nov 01 2021

Keywords

Links

Crossrefs

Cf. A190835, A348821, A348823; A007474, A348814, A348816, A348819.

A278991 a(n) is the number of simple linear diagrams with n+1 chords.

Original entry on oeis.org

0, 1, 3, 24, 211, 2325, 30198, 452809, 7695777, 146193678, 3069668575, 70595504859, 1764755571192, 47645601726541, 1381657584006399, 42829752879449400, 1413337528735664887, 49465522112961344241, 1830184115528550306438, 71375848864779552073957
Offset: 0

Views

Author

N. J. A. Sloane, Dec 07 2016

Keywords

Links

Crossrefs

Cf. A003436, A003437, A007474, A278990, A278992, A278993, A278994.

Programs

  • Mathematica
    a[0] = 0; a[1] = 1; a[2] = 3; a[n_] := a[n] = (2 n - 1) a[n - 1] + (4 n - 3) a[n - 2] + (2 n - 4) a[n - 3]; Table[a@ n, {n, 0, 19}] (* Michael De Vlieger, Dec 10 2016 *)
  • PARI
    seq(N) = {
  my(a = vector(N)); a[1]=1; a[2]=3; a[3]=24;
  for (n=4, N, a[n] = (2*n-1)*a[n-1] + (4*n-3)*a[n-2] + (2*n-4)*a[n-3]);
  concat(0, a);
};
seq(20) \\ Gheorghe Coserea, Dec 10 2016
  • PARI
    N = 20; x = 'x + O('x^N);
concat(0, Vec(serlaplace((1-sqrt(1-2*x))*(1-2*x)^(-3/2)*exp(-1-x+sqrt(1-2*x))))) \\ Gheorghe Coserea, Dec 10 2016

Formula

E.g.f.: (1-sqrt(1-2*x))*(1-2*x)^(-3/2)*exp(-1-x+sqrt(1-2*x)).
a(n) ~ 2^(n+3/2) * n^(n+1) / exp(n+3/2). - Vaclav Kotesovec, Dec 07 2016
a(n) = (2*n-1)*a(n-1) + (4*n-3)*a(n-2) + (2*n-4)*a(n-3). - Gheorghe Coserea, Dec 10 2016

Extensions

Offset corrected by Gheorghe Coserea, Dec 10 2016

A348814 a(n) = number of loopless diagrams by number of K_3 up to rotational symmetry.

Original entry on oeis.org

0, 1, 4, 126, 9367, 1120780, 189565588, 43154533233, 12735808866899, 4732638168795171, 2163220895025390670, 1193176166690983987122, 781607533669746761791541
Offset: 1

Views

Author

Michael De Vlieger, Nov 01 2021

Keywords

Links

Crossrefs

Cf. A190826, A292409, A348813; A007474, A348816, A348819, A348822.

A278992 Number of simple chord-labeled chord diagrams with n chords.

Original entry on oeis.org

0, 1, 1, 21, 168, 1968, 26094, 398653, 6872377, 132050271, 2798695656, 64866063276, 1632224748984, 44316286165297, 1291392786926821, 40202651019430461, 1331640833909877144, 46762037794122159492, 1735328399106396110310, 67858430028772637693845
Offset: 1

Views

Author

N. J. A. Sloane, Dec 07 2016

Keywords

Links

Crossrefs

Cf. A003436, A003437, A007474, A278990, A278991, A278993, A278994.

Programs

  • Mathematica
    terms = 20;
CoefficientList[(Sqrt[1 - 2t]+1)(1/Sqrt[1 - 2t])*E^(Sqrt[1 - 2t] - t - 1) - (2-t)/E^t + O[t]^(terms+1), t]*Range[0, terms]! // Rest (* Jean-François Alcover, Sep 14 2018 *)

Formula

E.g.f.: (1+sqrt(1-2*t))*(1-2*t)^(-1/2)*exp(-1-t+sqrt(1-2*t))-(2-t)*exp(-t).
a(n) ~ 2^(n+1/2) * n^n / exp(n+3/2). - Vaclav Kotesovec, Dec 07 2016
Conjecture D-finite with recurrence: +(-n+2)*a(n) +(2*n^2-8*n+7)*a(n-1) +(6*n^2-18*n+11)*a(n-2) +(n-1)*(6*n-11)*a(n-3) +2*(n-1)*(n-2)*a(n-4)=0. - R. J. Mathar, Jan 27 2020

A278993 Number of simple chord diagrams with n chords, up to rotation.

Original entry on oeis.org

0, 1, 1, 4, 21, 176, 1893, 25030, 382272, 6604535, 127222636, 2702798537, 62778105236, 1582725739329, 43046433007765, 1256332883208474, 39165907107963273, 1298945495674093932, 45666536827274985585, 1696460750775267473762
Offset: 1

Views

Author

N. J. A. Sloane, Dec 07 2016

Keywords

Links

Crossrefs

Cf. A003436, A003437, A007474, A278990, A278991, A278992, A278994.

A278994 Number of simple chord diagrams with n chords, modulo all symmetries.

Original entry on oeis.org

0, 1, 1, 4, 18, 116, 1060, 13019, 193425, 3313522, 63667788, 1351700744, 31390695708, 791372281393, 21523271532811, 628166776833181, 19582955637428422, 649472761243051940, 22833268501579122332, 848230375982060558217
Offset: 1

Views

Author

N. J. A. Sloane, Dec 07 2016

Keywords

Links

Crossrefs

Cf. A003436, A003437, A007474, A278990, A278991, A278992, A278993.
Showing 1-9 of 9 results.

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

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