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 1-7 of 7 results.

A290708 Number of irredundant sets in the (2 n)-crossed prism graph.

Original entry on oeis.org

9, 77, 609, 5221, 44914, 383279, 3267224, 27844845, 237311925, 2022585862, 17238412156, 146922412111, 1252214688188, 10672581690652, 90962034893584, 775266190295021, 6607566186841867, 56316051804768503, 479979708366734131, 4090849998653543166
Offset: 1

Views

Author

Eric W. Weisstein, Aug 09 2017

Keywords

Links

Crossrefs

Cf. A287062.

Formula

From Andrew Howroyd, Aug 10 2017: (Start)
Empirical: a(n) = 9*a(n-1) - 2*a(n-2) - 22*a(n-3) + 23*a(n-4) + 126*a(n-5) - 2*a(n-6) - 184*a(n-7) - 64*a(n-8) for n > 8.
Empirical g.f.: x*(9 - 4*x - 66*x^2 + 92*x^3 + 630*x^4 - 12*x^5 - 1288*x^6 - 512*x^7)/(1 - 9*x + 2*x^2 + 22*x^3 - 23*x^4 - 126*x^5 + 2*x^6 + 184*x^7 + 64*x^8).
(End)

Extensions

a(1) and terms a(7) and beyond from Andrew Howroyd, Aug 10 2017

A291772 Number of minimal dominating sets in the 2n-crossed prism graph.

Original entry on oeis.org

4, 12, 61, 316, 1304, 5223, 21557, 90404, 377863, 1572942, 6545785, 27262279, 113572619, 473082153, 1970443556, 8207168564, 34184621296, 142386794787, 593071821262, 2470268797246, 10289192009129, 42856677944829, 178507203892808, 743520516941183
Offset: 1

Views

Author

Eric W. Weisstein, Aug 31 2017

Keywords

Links

Crossrefs

Cf. A287062, A290708.

Programs

  • Mathematica
    Rest@ CoefficientList[Series[x (4 - 4 x + 21 x^2 + 68 x^3 + 10 x^4)/(1 - 4 x + 2 x^2 - 7 x^3 - 17 x^4 - 2 x^5), {x, 0, 24}], x] (* Michael De Vlieger, Aug 31 2017 *)
LinearRecurrence[{4,-2,7,17,2},{4,12,61,316,1304},30] (* Harvey P. Dale, Jul 02 2019 *)
Table[RootSum[-2 - 17 # - 7 #^2 + 2 #^3 - 4 #^4 + #^5 &, #^n &], {n, 20}] (* Eric W. Weisstein, Sep 08 2021 *)
  • PARI
    Vec((4 - 4*x + 21*x^2 + 68*x^3 + 10*x^4)/(1 - 4*x + 2*x^2 - 7*x^3 - 17*x^4 - 2*x^5)+O(x^30)) \\ Andrew Howroyd, Aug 31 2017
  • PARI
    \\ sequence prepended by a 5:
polsym(-2 - 17*x - 7*x^2 + 2*x^3 - 4*x^4 + x^5, 24) \\ Joerg Arndt, Sep 08 2021

Formula

From Andrew Howroyd, Aug 31 2017: (Start)
a(n) = 4*a(n-1) - 2*a(n-2) + 7*a(n-3) + 17*a(n-4) + 2*a(n-5) for n > 5.
G.f.: x*(4 - 4*x + 21*x^2 + 68*x^3 + 10*x^4)/(1 - 4*x + 2*x^2 - 7*x^3 - 17*x^4 - 2*x^5).
(End)

Extensions

a(1) and terms a(7) and beyond from Andrew Howroyd, Aug 31 2017

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

A287430 Number of connected dominating sets in the 2n-crossed prism graph.

Original entry on oeis.org

115, 1063, 9121, 75607, 611569, 4857223, 38034241, 294475447, 2258978449, 17196401383, 130059675361, 978211787287, 7322040929329, 54576195433543, 405286730532481, 2999780651211127, 22137879320864209, 162941058582753703, 1196418733436205601
Offset: 2

Views

Author

Eric W. Weisstein, May 25 2017

Keywords

Links

Crossrefs

Cf. A287062, A290757.

Programs

  • Mathematica
    Join[{115}, Table[7^(n - 3) (343 + 240 n), {n, 3, 20}]]
LinearRecurrence[{14, -49}, {115, 1063, 9121}, 19] (* amended by Georg Fischer, Apr 03 2019 *)
CoefficientList[Series[(115 - 547 x - 126 x^2)/(-1 + 7 x)^2, {x, 0, 20}], x]
  • PARI
    Vec((115 - 547*x - 126*x^2)/(1 - 7*x)^2 + O(x^20)) \\ Andrew Howroyd, Sep 05 2017

Formula

From Andrew Howroyd, Sep 05 2017: (Start)
a(n) = 7^n + 240*n*7^(n-3) for n > 2.
a(n) = 14*a(n-1) - 49*a(n-2) for n > 4.
G.f.: x^2*(115 - 547*x - 126*x^2)/(1 - 7*x)^2.
(End)

Extensions

Terms a(6) and beyond from Andrew Howroyd, Sep 05 2017

A287327 Number of independent vertex sets (and vertex covers) in the 2n-crossed prism graph.

Original entry on oeis.org

2, 7, 35, 196, 1127, 6517, 37730, 218491, 1265327, 7327852, 42437675, 245768761, 1423317602, 8242841887, 47736669995, 276456796756, 1601040887327, 9272088633997, 53697334226690, 310976719148851, 1800955694455127, 10429852827143932, 60402279928821635
Offset: 0

Views

Author

Andrew Howroyd, Aug 31 2017

Keywords

Comments

Sequence extrapolated to n = 0 using recurrence.

Links

Crossrefs

Cf. A276225 (maximal independent vertex sets), A287062, A290708.

Programs

  • Mathematica
    CoefficientList[Series[(2 - 7 x)/(1 - 7 x + 7 x^2), {x, 0, 22}], x] (* Michael De Vlieger, Aug 31 2017 *)
Table[(1/2 (7 - Sqrt[21]))^n + (1/2 (7 + Sqrt[21]))^n, {n, 0, 20}] // Expand (* Eric W. Weisstein, Sep 21 2017 *)
LinearRecurrence[{7, -7}, {7, 35}, {0, 20}] (* Eric W. Weisstein, Sep 21 2017 *)
  • PARI
    Vec((2 - 7*x)/(1 - 7*x + 7*x^2) + O(x^30))

Formula

a(n) = 7*a(n-1) - 7*a(n-2) for n > 1.
G.f.: (2 - 7*x)/(1 - 7*x + 7*x^2).

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 1-7 of 7 results.

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