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 1-3 of 3 results.

A124349 Numbers of directed Hamiltonian cycles on the n-prism graph.

Original entry on oeis.org

6, 12, 10, 16, 14, 20, 18, 24, 22, 28, 26, 32, 30, 36, 34, 40, 38, 44, 42, 48, 46, 52, 50, 56, 54, 60, 58, 64, 62, 68, 66, 72, 70, 76, 74, 80, 78, 84, 82, 88, 86, 92, 90, 96, 94, 100, 98, 104, 102, 108, 106, 112, 110, 116, 114, 120, 118, 124, 122, 128, 126, 132, 130
Offset: 3

Views

Author

Eric W. Weisstein, Oct 26 2006

Keywords

Links

Crossrefs

Cf. A014681, A124350, A270273.

Programs

  • Magma
    [2*n+(1-(n mod 2))*4: n in [3..80]]; // Vincenzo Librandi, Jan 26 2016
  • Maple
    seq( 2*n + (1-(n mod 2))*4, n=3..100); # Robert Israel, Mar 14 2016
  • Mathematica
    Table[2 n + (1 - Mod[n, 2]) 4, {n, 3, 100}] (* Vincenzo Librandi, Jan 26 2016 *)
  • PARI
    Vec(2*x^3*(3+3*x-4*x^2)/((1-x)^2*(1+x)) + O(x^100)) \\ Altug Alkan, Mar 14 2016

Formula

a(n) = 2*n + (1-(n mod 2))*4.
From Colin Barker, Aug 22 2012: (Start)
a(n) = a(n-1)+a(n-2)-a(n-3).
G.f.: 2*x^3*(3+3*x-4*x^2)/((1-x)^2*(1+x)). (End)
a(n) = 2*A014681(n+1). - R. J. Mathar, Jan 25 2016
E.g.f.: 2*(2 + x)*cosh(x) + 2*x*sinh(x) - 2*(2 + x + 2*x^2). - Stefano Spezia, Jan 28 2024

Extensions

Name clarified by Andrew Howroyd, Mar 14 2016

A287992 Number of (undirected) paths in the prism graph Y_n.

Original entry on oeis.org

1, 26, 129, 444, 1285, 3366, 8281, 19544, 44829, 100770, 223201, 488916, 1061749, 2289854, 4910505, 10480176, 22275661, 47178234, 99605809, 209704940, 440390181, 922733526, 1929364729, 4026514824, 8388588925, 17448283346, 36238762881, 75161901444, 155692535509, 322122515310
Offset: 1

Views

Author

Eric W. Weisstein, Jun 04 2017

Keywords

Comments

Extended to a(1)-a(2) using the formula.

Links

Crossrefs

Cf. A077265, A124350, A124349, A287988, A020874, A288516.

Programs

  • Mathematica
    Table[(5 2^(n + 1) - 5 n - n^2 - 13) n, {n, 20}]
LinearRecurrence[{8, -26, 44, -41, 20, -4}, {1, 26, 129, 444, 1285, 3366}, 20]
CoefficientList[Series[(1 + 18 x - 53 x^2 + 44 x^3 - 16 x^4)/((1 - x)^4 (1 - 2 x)^2), {x, 0, 20}], x]
  • PARI
    Vec(x*(1 + 18*x - 53*x^2 + 44*x^3 - 16*x^4) / ((1 - x)^4*(1 - 2*x)^2) + O(x^30)) \\ Colin Barker, Jun 04 2017

Formula

a(n) = (5*2^(n + 1) - 5*n - n^2 - 13)*n.
From Colin Barker, Jun 04 2017: (Start)
G.f.: x*(1 + 18*x - 53*x^2 + 44*x^3 - 16*x^4) / ((1 - x)^4*(1 - 2*x)^2).
a(n) = 8*a(n-1) - 26*a(n-2) + 44*a(n-3) - 41*a(n-4) + 20*a(n-5) - 4*a(n-6) for n>6. (End)

A308137 Number of (undirected) Hamiltonian paths on the n-prism graph.

Original entry on oeis.org

30, 72, 130, 228, 350, 528, 738, 1020, 1342, 1752, 2210, 2772, 3390, 4128, 4930, 5868, 6878, 8040, 9282, 10692, 12190, 13872, 15650, 17628, 19710, 22008, 24418, 27060, 29822, 32832, 35970, 39372, 42910, 46728, 50690, 54948, 59358, 64080, 68962, 74172, 79550
Offset: 3

Views

Author

Eric W. Weisstein, May 14 2019

Keywords

Links

Crossrefs

Cf. A124350.

Programs

  • Mathematica
    LinearRecurrence[{2,1,-4,1,2,-1},{30,72,130,228,350,528},50] (* Harvey P. Dale, Jun 20 2021 *)
  • PARI
    Vec(2*x^3*(15 + 6*x - 22*x^2 + 8*x^3 + 11*x^4 - 6*x^5) / ((1 - x)^4*(1 + x)^2) + O(x^30)) \\ Colin Barker, Jul 19 2019

Formula

a(n) = A124350(n)/2.
From Colin Barker, Jul 19 2019: (Start)
G.f.: 2*x^3*(15 + 6*x - 22*x^2 + 8*x^3 + 11*x^4 - 6*x^5) / ((1 - x)^4*(1 + x)^2).
a(n) = n*(3 + (-1)^n + 2*n^2) / 2.
a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6) for n>8.
(End)

Extensions

More terms from Colin Barker, Jul 19 2019
Showing 1-3 of 3 results.

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