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.

A302652 Number of minimum total dominating sets in the n-antiprism graph.

Original entry on oeis.org

2, 6, 12, 24, 80, 48, 7, 16, 237, 40, 154, 1344, 208, 7, 30, 1136, 68, 396, 6688, 480, 7, 44, 3151, 96, 750, 20800, 864, 7, 58, 6730, 124, 1216, 50160, 1360, 7, 72, 12321, 152, 1794, 103040, 1968, 7, 86, 20372, 180, 2484, 189504, 2688, 7, 100, 31331, 208, 3286
Offset: 1

Views

Author

Eric W. Weisstein, Apr 11 2018

Keywords

Comments

Sequence extrapolated to n=1 using recurrence.

Links

Crossrefs

Cf. A302255, A302760, A302763.

Programs

  • Mathematica
    Table[Piecewise[{{7, Mod[n, 7] == 0}, {2 n, Mod[n, 7] == 1}, {n (37 + 138 n + 32 n^2)/147, Mod[n, 7] == 2}, {4 n, Mod[n, 7] == 3}, {2 n (5 + 4 n)/7, Mod[n, 7] == 4}, {(8 n (2 + n) (9 + n) (1 + 4 n))/1029, Mod[n, 7] == 5}, {8 n (1 + n)/7, Mod[n, 7] == 6}}], {n, 200}]
LinearRecurrence[{0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, -10, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, -5, 0, 0, 0, 0, 0, 0, 1}, {2, 6, 12, 24, 80, 48, 7, 16, 237, 40, 154, 1344, 208, 7, 30, 1136, 68, 396, 6688, 480, 7, 44, 3151, 96, 750, 20800, 864, 7, 58, 6730, 124, 1216, 50160, 1360, 7}, 200]
Rest @ CoefficientList[Series[7 x^7/(1 - x^7) - 16 x^6 (3 + 4 x^7)/(-1 + x^7)^3 + 4 x^3 (3 + 4 x^7)/(-1 + x^7)^2 + 2 x (1 + 6 x^7)/(-1 + x^7)^2 - 2 x^4 (12 + 41 x^7 + 3 x^14)/(-1 + x^7)^3 - 16 x^5 (5 + 59 x^7 + 48 x^14)/(-1 + x^7)^5 + x^2 (6 + 213 x^7 + 224 x^14 + 5 x^21)/(-1 + x^7)^4, {x, 0, 200}], x]
  • PARI
    a(n)={[k->7, k->2*(7*k+1), k->(7*k+2)*(32*k^2+38*k+9)/3, k->4*(7*k+3), k->(7*k+4)*(8*k+6), k->(7*k+5)*(8*k+8)*(k+2)*(4*k+3)/3, k->8*(7*k+6)*(k+1)][1+n%7](n\7)} \\ Andrew Howroyd, Apr 18 2018

Formula

From Andrew Howroyd, Apr 18 2018: (Start)
a(n) = 5*a(n-7) - 10*a(n-14) + 10*a(n-21) - 5*a(n-28) + a(n-35).
a(7k) = 7, a(7k+1) = 2*(7*k+1), a(7k+2) = (7*k+2)*(32*k^2+38*k+9)/3, a(7k+3) = 4*(7*k+3), a(7k+4) = (7*k+4)*(8*k+6), a(7k+5) = (7*k+5)*(8*k+8)*(k+2)*(4*k+3)/3, a(7k+6) = 8*(7*k+6)*(k+1). (End)

Extensions

a(1)-a(2) and terms a(15) and beyond from Andrew Howroyd, Apr 18 2018

A302760 Number of total dominating sets in the n-antiprism graph.

Original entry on oeis.org

3, 11, 54, 179, 648, 2414, 8809, 32195, 117945, 431696, 1579955, 5783294, 21168592, 77482521, 283608249, 1038086883, 3799689944, 13907938601, 50906985592, 186333942984, 682034858839, 2496440225499, 9137676323347, 33446476209566, 122423549667123
Offset: 1

Views

Author

Eric W. Weisstein, Apr 12 2018

Keywords

Comments

Sequence extrapolated to n=1 using recurrence. - Andrew Howroyd, Apr 14 2018

Links

Crossrefs

Cf. A302255, A302652, A302763.

Programs

  • Magma
    I:=[3,11,54,179,648,2414,8809]; [n le 7 select I[n] else 3*Self(n-1)+Self(n-2)+6*Self(n-3)-3*Self(n-4)+Self(n-7): n in [1..30]]; // Vincenzo Librandi, Apr 15 2018
  • Mathematica
    CoefficientList[Series[(3 + 2 x + 18 x^2 - 12 x^3 + 7 x^6)/(1 - 3 x - x^2 - 6 x^3 + 3 x^4 - x^7), {x, 0, 24}], x] (* Michael De Vlieger, Apr 14 2018 *)
LinearRecurrence[{3, 1, 6, -3, 0, 0, 1}, {3, 11, 54, 179, 648, 2414, 8809}, 20] (* Vincenzo Librandi, Apr 15 2018 *)
Table[RootSum[-1 + 3 #^3 - 6 #^4 - #^5 - 3 #^6 + #^7 &, #^n &], {n, 30}] (* Eric W. Weisstein, Apr 16 2018 *)
RootSum[-1 + 3 #^3 - 6 #^4 - #^5 - 3 #^6 + #^7 &, #^Range[30] &] (* Eric W. Weisstein, Apr 16 2018 *)
  • PARI
    Vec((3 + 2*x + 18*x^2 - 12*x^3 + 7*x^6)/(1 - 3*x - x^2 - 6*x^3 + 3*x^4 - x^7) + O(x^25)) \\ Andrew Howroyd, Apr 14 2018

Formula

From Andrew Howroyd, Apr 14 2018: (Start)
a(n) = 3*a(n-1) + a(n-2) + 6*a(n-3) - 3*a(n-4) + a(n-7) for n > 7.
G.f.: x*(3 + 2*x + 18*x^2 - 12*x^3 + 7*x^6)/(1 - 3*x - x^2 - 6*x^3 + 3*x^4 - x^7).
(End)

Extensions

a(1)-a(2) and terms a(11) and beyond from Andrew Howroyd, Apr 14 2018

A302255 Total domination number of the n-antiprism graph.

Original entry on oeis.org

0, 1, 2, 2, 3, 4, 4, 4, 5, 6, 6, 7, 8, 8, 8, 9, 10, 10, 11, 12, 12, 12, 13, 14, 14, 15, 16, 16, 16, 17, 18, 18, 19, 20, 20, 20, 21, 22, 22, 23, 24, 24, 24, 25, 26, 26, 27, 28, 28, 28, 29, 30, 30, 31, 32, 32, 32, 33, 34, 34, 35, 36, 36, 36, 37, 38, 38, 39, 40, 40, 40, 41
Offset: 0

Views

Author

Eric W. Weisstein, Apr 07 2018

Keywords

Comments

Sequence extended to a(0)-a(2) using the recurrence/formula.

Links

Crossrefs

Cf. A302652, A302760, A302763.

Programs

  • Magma
    I:=[2,2,3,4,4,4,5,6]; [0,1] cat [n le 8 select I[n] else Self(n-1) + Self(n-7) - Self(n-8): n in [1..30]]; // G. C. Greubel, Apr 09 2018
  • Mathematica
    Table[(4 + 4 n + E^(4 I n Pi/7) Root[1 + 7 #^2 + 28 #^3 + 42 #^4 + 28 #^5 + 7 #^6 &, 1] + E^(-4 I n Pi/7) Root[1 + 7 #^2 + 28 #^3 + 42 #^4 + 28 #^5 + 7 #^6 &, 2] + E^(-2 I n Pi/7) Root[1 + 7 #^2 + 28 #^3 + 42 #^4 + 28 #^5 + 7 #^6 &, 3] + E^(2 I n Pi/7) Root[1 + 7 #^2 + 28 #^3 + 42 #^4 + 28 #^5 + 7 #^6 &, 4] + E^(-6 I n Pi/7) Root[1 + 7 #^2 + 28 #^3 + 42 #^4 + 28 #^5 + 7 #^6 &, 5] + E^(6 I n Pi/7) Root[1 + 7 #^2 + 28 #^3 + 42 #^4 + 28 #^5 + 7 #^6 &, 6])/ 7, {n, 20}] // RootReduce
LinearRecurrence[{1,0,0,0,0,0,1,-1}, {1,2,2,3,4,4,4,5,6,6}, {0, 20}]
CoefficientList[Series[x (1 + x + x^3 + x^4)/((1 - x)^2 (1 + x + x^2 + x^3 + x^4 + x^5 + x^6)), {x, 0, 20}], x]
  • PARI
    x='x+O('x^50); concat(0, Vec(x*(1+x+x^3+x^4)/((1-x)^2*(1+x+x^2+ x^3+x^4+x^5+x^6)))) \\ G. C. Greubel, Apr 09 2018

Formula

a(n) = a(n-1) + a(n-7) - a(n-8).
G.f.: x*(1 + x + x^3 + x^4)/((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)).
a(n) = a(n-7) + 4. - Andrew Howroyd, Apr 18 2018
a(n) = a(n-7*k) + 4*k. - Eric W. Weisstein, Apr 19 2018
Showing 1-3 of 3 results.

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