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.

A362750 Number of total dominating sets in the n-double cone graph.

Original entry on oeis.org

4, 16, 79, 336, 1144, 4351, 17224, 67936, 267919, 1063216, 4233904, 16882191, 67380304, 269142736, 1075602319, 4299846976, 17192621224, 68752838911, 274965310744, 1099740514416, 4398645585679, 17593754283616, 70372850295904, 281485727082511, 1125928050595744
Offset: 1

Views

Author

Eric W. Weisstein, May 02 2023

Keywords

Comments

The n-double cone graph is defined for n >= 3. The sequence has been extended to n=1 using the formula/recurrence. - Andrew Howroyd, May 03 2023

Links

Crossrefs

Cf. A000032, A001638, A056594, A302603.

Programs

  • Mathematica
    Table[1 + 4 (-1)^n + 4^n + LucasL[2 n] + 4 LucasL[n] Cos[n Pi/2], {n, 20}] (* Eric W. Weisstein, Sep 09 2023 *)
Table[(2 Cos[n Pi/2] + Fibonacci[n + 1] + Fibonacci[n - 1])^2 + 4^n - 1, {n, 20}] (* Eric W. Weisstein, Sep 09 2023 *)
LinearRecurrence[{7, -15, 18, -24, -6, 27, -15, 13, -4}, {4, 16, 79, 336, 1144, 4351, 17224, 67936, 267919}, 20] (* Eric W. Weisstein, Sep 09 2023 *)
CoefficientList[Series[4/(1 - 4 x) + 1/(1 - x) - 4/(1 + x) + (3 - 2 x)/(1 + (-3 + x) x) - 4 x (3 + 2 x^2)/(1 + 3 x^2 + x^4), {x, 0, 20}], x] (* Eric W. Weisstein, Sep 09 2023 *)
  • PARI
    a(n) = {(fibonacci(n+1) + fibonacci(n-1) + I^n + (-I)^n)^2 + 4^n - 1} \\ Andrew Howroyd, May 03 2023

Formula

From Andrew Howroyd, May 03 2023: (Start)
a(n) = A001638(n)^2 + 4^n - 1.
a(n) = (A000032(n) + 2*A056594(n))^2 + 4^n - 1.
a(2*n-1) = A302603(4*n-1).
a(n) = 7*a(n-1) - 15*a(n-2) + 18*a(n-3) - 24*a(n-4) - 6*a(n-5) + 27*a(n-6) - 15*a(n-7) + 13*a(n-8) - 4*a(n-9) for n > 9.
G.f.: x*(4 - 12*x + 27*x^2 - 49*x^3 - 215*x^4 + 369*x^5 - 237*x^6 + 207*x^7 - 64*x^8)/((1 - x)*(1 + x)*(1 - 4*x)*(1 - 3*x + x^2)*(1 + 3*x^2 + x^4)).
(End)

Extensions

a(1)-a(2) prepended and a(16) and beyond from Andrew Howroyd, May 03 2023

A302762 Number of minimal total dominating sets in the n-Andrásfai graph.

Original entry on oeis.org

1, 5, 14, 44, 112, 238, 449, 782, 1287, 2030, 3096, 4592, 6650, 9430, 13123, 17954, 24185, 32118, 42098, 54516, 69812, 88478, 111061, 138166, 170459, 208670, 253596, 306104, 367134, 437702, 518903, 611914, 717997, 838502, 974870, 1128636, 1301432, 1494990, 1711145
Offset: 1

Views

Author

Eric W. Weisstein, Apr 12 2018

Keywords

Links

Crossrefs

Cf. A213661, A290270, A302603.

Programs

  • Mathematica
    Join[{1, 5}, Table[(-720 + 2732 n - 1880 n^2 + 505 n^3 - 40 n^4 + 3 n^5)/120, {n, 3, 20}]]
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 5, 14, 44, 112, 238, 449, 782}, 20]
CoefficientList[Series[(1 - x - x^2 + 15 x^3 - 27 x^4 + 15 x^5 + 2 x^6 - x^7)/(-1 + x)^6, {x, 0, 20}], x]

Formula

a(n) = (-720 + 2732*n - 1880*n^2 + 505*n^3 - 40*n^4 + 3*n^5)/120 for n > 2.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n > 8.
G.f.: x*(1 - x - x^2 + 15*x^3 - 27*x^4 + 15*x^5 + 2*x^6 - x^7)/(-1 + x)^6.

Extensions

a(8)-a(20) from Andrew Howroyd, Apr 15 2018

A302658 Number of minimal total dominating sets in the wheel graph on n nodes.

Original entry on oeis.org

1, 2, 6, 8, 10, 15, 14, 12, 21, 35, 33, 37, 52, 63, 83, 116, 136, 162, 228, 309, 388, 506, 667, 865, 1155, 1547, 2010, 2629, 3509, 4654, 6138, 8132, 10750, 14195, 18842, 25000, 33041, 43719, 57957, 76769, 101680, 134731, 178407, 236240, 313052, 414782, 549336
Offset: 2

Views

Author

Eric W. Weisstein, Apr 11 2018

Keywords

Comments

Wheel graphs are defined for n>=4; extended to n=2 using formula. - Andrew Howroyd, Apr 15 2018

Links

Crossrefs

Cf. A213661, A290270, A300738, A302603.

Programs

  • Mathematica
    Table[n - 1 + RootSum[-1 - # + #^3 &, #^(n - 1) &] + (1 - (-1)^n) RootSum[-1 + #^2 + #^3 &, #^((n - 1)/2) &], {n, 2, 50}]
LinearRecurrence[{2, -1, 1, -1, 0, 0, -1, 0, 1, 1, -1}, {1, 2, 6, 8, 10, 15, 14, 12, 21, 35, 33}, 50]
CoefficientList[Series[(1 + 3 x^2 - 3 x^3 - x^4 - x^5 - 8 x^6 - 2 x^7 + 8 x^8 + 11 x^9 - 9 x^10)/((-1 + x)^2 (1 - x^3 - x^4 - x^5 - x^6 + x^8 + x^9)), {x, 0, 50}], x]
  • PARI
    {my(v=concat([0,0],Vec((3 + 4*x + 5*x^2 + 6*x^3 - 8*x^5 - 9*x^6)/((1 - x^2 - x^3)*(1 + x^2 - x^6)) + O(x^50))));vector(#v,i,v[i]+i)} \\ Andrew Howroyd, Apr 15 2018

Formula

a(n) = A300738(n-1) + (n-1). - Andrew Howroyd, Apr 15 2018
G.f.: x^2*(1 + 3*x^2 - 3*x^3 - x^4 - x^5 - 8*x^6 - 2*x^7 + 8*x^8 + 11*x^9 - 9*x^10)/((-1 + x)^2*(1 - x^3 - x^4 - x^5 - x^6 + x^8 + x^9)).

Extensions

a(2)-a(3) and terms a(20) and beyond from Andrew Howroyd, Apr 15 2018
Showing 1-3 of 3 results.

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