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.

A077263 Number of (undirected) cycles in the n-th order antiprism graph.

Original entry on oeis.org

63, 179, 523, 1619, 5239, 17379, 58323, 196691, 664623, 2247443, 7601883, 25715603, 86993639, 294295491, 995592355, 3368062355, 11394070559, 38545861491, 130399711915, 441139061715, 1492362751831, 5048627021731, 17079382870643, 57779138376659
Offset: 3

Views

Author

Eric W. Weisstein, Nov 01 2002

Keywords

Comments

Also the number of minimal edge cuts in the n-trapezohedron graph. - Eric W. Weisstein, Dec 11 2024

Links

Crossrefs

Cf. A077265.

Programs

  • Mathematica
    LinearRecurrence[{6, -11, 8, -3, 2, -1}, {63, 179, 523, 1619, 5239, 17379}, 22] (* Eric W. Weisstein, Dec 19 2013 *)
Table[4 n (n - 1) + RootSum[-1 - # - 3 #^2 + #^3 &, #^n &], {n, 3, 20}] (* Eric W. Weisstein, May 05 2017 *)
CoefficientList[Series[(63 - 199 x + 142 x^2 - 54 x^3 + 35 x^4 - 19 x^5)/((-1 + x)^3 (-1 + 3 x + x^2 + x^3)), {x, 0, 20}], x] (* Eric W. Weisstein, Dec 14 2017 *)

Formula

a(n) = 6*a(n-1) - 11*a(n-2) + 8*a(n-3) - 3*a(n-4) + 2*a(n-5) - a(n-6) for n>8. - Eric W. Weisstein, Dec 19 2013
G.f.: x^3*(63 - 199*x + 142*x^2 - 54*x^3 + 35*x^4 - 19*x^5)/((1 - x)^3*(1 - 3*x - x^2 - x^3)). - Bruno Berselli, Dec 20 2013

Extensions

a(7)-a(10) from Max Alekseyev, May 02 2010

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)

A290699 a(n) = 2^n - n + n^2.

Original entry on oeis.org

0, 2, 6, 14, 28, 52, 94, 170, 312, 584, 1114, 2158, 4228, 8348, 16566, 32978, 65776, 131344, 262450, 524630, 1048956, 2097572, 4194766, 8389114, 16777768, 33555032, 67109514, 134218430, 268436212, 536871724, 1073742694, 2147484578, 4294968288, 8589935648, 17179870306
Offset: 0

Views

Author

Eric W. Weisstein, Aug 09 2017

Keywords

Comments

Number of minimal edge covers in the n-book graph.

Links

Crossrefs

Cf. A132109.
A077265 is a subsequence.

Programs

  • Mathematica
    Table[2^n - n + n^2, {n, 20}]
LinearRecurrence[{5, -9, 7, -2}, {2, 6, 14, 28}, 20]
CoefficientList[Series[-((2 (-1 + 2 x - x^2 + x^3))/((-1 + x)^3 (-1 + 2 x))), {x, 0, 20}], x]

Formula

a(n) = 2*A132109(n-1).
a(n) = 5*a(n-1) - 9*a(n-2) + 7*a(n-3) - 2*a(n-4).
G.f.: -((2 x (-1 + 2 x - x^2 + x^3))/((-1 + x)^3 (-1 + 2 x))).

Extensions

Added a(0)=0. - N. J. A. Sloane, May 25 2019
Showing 1-3 of 3 results.

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