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 1-4 of 4 results.

A270273 Array read by antidiagonals: T(n,m) = number of Hamiltonian cycles in C_n X C_m.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 3, 3, 1, 1, 6, 48, 6, 1, 1, 5, 126, 126, 5, 1, 1, 8, 390, 1344, 390, 8, 1, 1, 7, 1014, 2930, 2930, 1014, 7, 1, 1, 10, 2982, 28060, 23580, 28060, 2982, 10, 1, 1, 9, 8094, 55230, 145210, 145210, 55230, 8094, 9, 1
Offset: 1

Views

Author

Andrew Howroyd, Mar 14 2016

Keywords

Examples

			The start of the sequence as table:
  0 0    1     1       1        1         1 ...
  0 1    3     6       5        8         7 ...
  1 3   48   126     390     1014      2982 ...
  1 6  126  1344    2930    28060     55230 ...
  1 5  390  2930   23580   145210   1045940 ...
  1 8 1014 28060  145210  3273360  16111928 ...
  1 7 2982 55230 1045940 16111928 257165468 ...
  ...

Links

Crossrefs

Row n=3-5 give: A194952, A216588, A358853.
Main diagonal gives A222199.

Formula

T(n,2) = A124349(n) / 2.

A124350 a(n) = 4*n*(floor(n^2/2)+1). For n >= 3, this is the number of directed Hamiltonian paths on the n-prism graph.

Original entry on oeis.org

0, 4, 24, 60, 144, 260, 456, 700, 1056, 1476, 2040, 2684, 3504, 4420, 5544, 6780, 8256, 9860, 11736, 13756, 16080, 18564, 21384, 24380, 27744, 31300, 35256, 39420, 44016, 48836, 54120, 59644, 65664, 71940, 78744, 85820, 93456, 101380, 109896, 118716
Offset: 0

Views

Author

Eric W. Weisstein, Oct 26 2006

Keywords

Links

Crossrefs

Cf. A080827, A124349.

Programs

  • Mathematica
    LinearRecurrence[{2, 1, -4, 1, 2, -1}, {0, 4, 24, 60, 144, 260}, 60] (* Vincenzo Librandi, Jan 26 2016 *)
  • PARI
    Vec(4*x*(x^2+1)*(x^2+4*x+1)/((x-1)^4*(x+1)^2) + O(x^100)) \\ Colin Barker, Sep 06 2013

Formula

From Colin Barker, Sep 06 2013: (Start)
a(n) = n*(3 + (-1)^n + 2*n^2).
G.f.: 4*x*(x^2+1)*(x^2+4*x+1) / ((x-1)^4*(x+1)^2). (End)
a(n) = 4*n*A080827(n). - R. J. Mathar, Jan 25 2016
E.g.f.: 2*x*((2 + 3*x + x^2)*cosh(x) + (3 + 3*x + x^2)*sinh(x)). - Stefano Spezia, Jan 27 2024

Extensions

Formula and further terms from Max Alekseyev, Feb 07 2008

A338153 a(n) is the number of acyclic orientations of the edges of the n-prism.

Original entry on oeis.org

204, 1862, 14700, 109334, 790524, 5633222, 39828300, 280376054, 1968934044, 13807724582, 96754776300, 677686169174, 4745413960764, 33224340503942, 232596153986700, 1628276158432694, 11398345428510684, 79790067272259302, 558537067986067500, 3909785864202510614
Offset: 3

Views

Author

Peter Kagey, Oct 13 2020

Keywords

Comments

Conjectured linear recurrence and g.f. confirmed by Kagey's formula. - Ray Chandler, Mar 10 2024

Examples

			For n = 4, the 4-prism is the 3-dimensional cube, so a(4) = A334247(3) = 1862.

Links

Crossrefs

Cf. A102080, A124349, A284702.
Cf. A033815 (cross-polytope), A058809 (wheel), A334247 (cube), A338152 (n-demihypercube), A338154 (n-antiprism).

Programs

Formula

Conjectures from Colin Barker, Oct 13 2020: (Start)
G.f.: 2*x^3*(102 - 497*x + 742*x^2 - 392*x^3) / ((1 - x)*(1 - 2*x)*(1 - 4*x)*(1 - 7*x)).
a(n) = 14*a(n-1) - 63*a(n-2) + 106*a(n-3) - 56*a(n-4) for n>6.
(End)
a(n) = 5 + 7^n - 2^(n+1) - 2*4^n. - Peter Kagey, Nov 15 2020

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)
Showing 1-4 of 4 results.

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