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.

A302946 Number of minimal (and minimum) total dominating sets in the 2n-crossed prism graph.

Original entry on oeis.org

4, 36, 196, 1156, 6724, 39204, 228484, 1331716, 7761796, 45239076, 263672644, 1536796804, 8957108164, 52205852196, 304278004996, 1773462177796, 10336495061764, 60245508192804, 351136554095044, 2046573816377476, 11928306344169796, 69523264248641316
Offset: 1

Views

Author

Eric W. Weisstein, Apr 16 2018

Keywords

Comments

Extended to a(1) using the formula/recurrence.
Since minimal and minimum total dominating sets are equivalent, the crossed prism graphs could be said to be "well totally dominated".

Links

Crossrefs

Cf. A001333, A002203 (sqrt), A090390 (quarter), A287062, A291772, A302941.

Programs

  • Mathematica
    Table[2 (ChebyshevT[n, 3] + (-1)^n), {n, 20}]
Table[4 (-1)^n ChebyshevT[n, I]^2, {n, 20}]
LinearRecurrence[{5, 5, -1}, {4, 36, 196}, 20]
CoefficientList[Series[-4 (-1 - 4 x + x^2)/(1 - 5 x - 5 x^2 + x^3), {x, 0, 20}], x]
  • PARI
    Vec(4*(1 + 4*x - x^2)/((1 + x)*(1 - 6*x + x^2)) + O(x^30)) \\ Andrew Howroyd, Apr 16 2018
  • PARI
    a(n) = 2*(polchebyshev(n,1,3) + (-1)^n); \\ Michel Marcus, Apr 17 2018

Formula

From Andrew Howroyd, Apr 16 2018: (Start)
a(n) = 5*a(n-1) + 5*a(n-2) - a(n-3).
G.f.: 4*x*(1 + 4*x - x^2)/((1 + x)*(1 - 6*x + x^2)).
a(n) = 4*A090390(n) = 4*A001333(n)^2. (End)
a(n) = 2*(chebyshevT(n,3) + (-1)^n). - Eric W. Weisstein, Apr 17 2018
a(n) = 4*(-1)^n*chebyshevT(n,i)^2, where i is the imaginary unit. - Eric W. Weisstein, Apr 17 2018
E.g.f.: 2*(exp(-x) + exp(3*x)*cosh(2*sqrt(2)*x) - 2). - Stefano Spezia, Aug 03 2024

Extensions

a(1) and terms a(6) and beyond from Andrew Howroyd, Apr 16 2018

A302941 Number of total dominating sets in the 2n-crossed prism graph.

Original entry on oeis.org

9, 121, 1296, 14161, 154449, 1684804, 18378369, 200477281, 2186871696, 23855111401, 260219353689, 2838557779204, 30963916217529, 337764520613641, 3684445810532496, 40191139395243841, 438418087537149729, 4782407823513403204, 52168067971110285489
Offset: 1

Views

Author

Eric W. Weisstein, Apr 16 2018

Keywords

Links

Crossrefs

Cf. A006497, A287062, A291772, A302946.

Programs

  • Mathematica
    Table[2 (-1)^n + ((11 - 3 Sqrt[13])/2)^n + ((11 + 3 Sqrt[13])/2)^n, {n, 20}] // FullSimplify
Table[LucasL[n, 3]^2, {n, 20}]
LucasL[Range[20], 3]^2
LinearRecurrence[{10, 10, -1}, {9, 121, 1296}, 20]
CoefficientList[Series[(9 + 31 x - 4 x^2)/(1 - 10 x - 10 x^2 + x^3), {x, 0, 20}], x]
  • PARI
    Vec((9 + 31*x - 4*x^2)/((1 + x)*(1 - 11*x + x^2)) + O(x^30)) \\ Andrew Howroyd, Apr 16 2018

Formula

From Andrew Howroyd, Apr 16 2018: (Start)
G.f.: x*(9 + 31*x - 4*x^2)/((1 + x)*(1 - 11*x + x^2)).
a(n) = 10*a(n-1) + 10*a(n-2) - a(n-3) for n > 3.
a(n) = A006497(n)^2. (End)

Extensions

a(1) and terms a(6) and beyond from Andrew Howroyd, Apr 16 2018

A347551 Number of minimum dominating sets in the 2n-crossed prism graph.

Original entry on oeis.org

4, 51, 8, 170, 16, 476, 32, 1224, 64, 2992, 128, 7072, 256, 16320, 512, 36992, 1024, 82688, 2048, 182784, 4096, 400384, 8192, 870400, 16384, 1880064, 32768, 4038656, 65536, 8634368, 131072, 18382848, 262144, 38993920, 524288, 82444288, 1048576, 173801472
Offset: 2

Views

Author

Eric W. Weisstein, Sep 06 2021

Keywords

Links

Crossrefs

Cf. A287062, A291772, A302946.

Programs

  • Mathematica
    Table[Piecewise[{{17 n 2^((n - 3)/2), Mod[n, 2] == 1}, {2^((n/2) + 1), Mod[n, 2] == 0}}], {n, 2, 20}] (* Eric W. Weisstein, Feb 27 2025 *)
CoefficientList[Series[(4 + 51 x - 8 x^2 - 34 x^3)/(1 - 2 x^2)^2, {x, 0, 20}], x] (* Eric W. Weisstein, Feb 27 2025 *)
  • PARI
    a(n) = if(n%2, 17*n*2^((n-3)/2), 2^((n/2)+1)) \\ Andrew Howroyd, Jan 18 2022

Formula

a(n) = 2^((n/2)+1) for n even.
From Andrew Howroyd, Jan 18 2022: (Start)
a(n) = 17*n*2^((n-3)/2) for n odd.
a(n) = 4*a(n-2) - 4*a(n-4) for n > 5.
G.f.: x^2*(4 + 51*x - 8*x^2 - 34*x^3)/(1 - 2*x^2)^2.
(End)

Extensions

Terms a(11) and beyond from Andrew Howroyd, Jan 18 2022
Showing 1-3 of 3 results.

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