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 1-4 of 4 results.

A296102 Number of total dominating sets in the n-prism graph.

Original entry on oeis.org

3, 9, 39, 121, 443, 1521, 5071, 17161, 58035, 196249, 664183, 2247001, 7601259, 25715041, 86992799, 294294025, 995591267, 3368061225, 11394069191, 38545861561, 130399710235, 441139057489, 1492362749807, 5048627017225, 17079382868243, 57779138385081, 195465425009943
Offset: 1

Views

Author

Eric W. Weisstein, Apr 16 2018

Keywords

Comments

Sequence extrapolated to n=1 using recurrence. - Andrew Howroyd, Apr 16 2018

Links

Crossrefs

Cf. A284702, A290336, A295420.

Programs

  • Mathematica
    Table[RootSum[-1 - # - 3 #^2 + #^3 &, #^n &] + RootSum[1 + # - #^2 + #^3 &, #^n &] + RootSum[-1 + # + #^2 + #^3 &, #^n &], {n, 20}]
LinearRecurrence[{3, 0, 4, -2, 10, 4, 0, -1, -1}, {3, 9, 39, 121, 443,
   1521, 5071, 17161, 58035}, 20]
CoefficientList[Series[(3 + 12 x^2 - 8 x^3 + 50 x^4 + 24 x^5 - 8 x^7 - 9 x^8)/(1 - 3 x - 4 x^3 + 2 x^4 - 10 x^5 - 4 x^6 + x^8 + x^9), {x, 0, 20}], x]
  • PARI
    Vec((3 + 12*x^2 - 8*x^3 + 50*x^4 + 24*x^5 - 8*x^7 - 9*x^8)/((1 - x + x^2 + x^3)*(1 + x + x^2 - x^3)*(1 - 3*x - x^2 - x^3)) + O(x^30)) \\ Andrew Howroyd, Apr 16 2018

Formula

From Andrew Howroyd, Apr 16 2018: (Start)
G.f.: x*(3 + 12*x^2 - 8*x^3 + 50*x^4 + 24*x^5 - 8*x^7 - 9*x^8)/((1 - x + x^2 + x^3)*(1 + x + x^2 - x^3)*(1 - 3*x - x^2 - x^3)).
a(n) = 3*a(n-1) + 4*a(n-3) - 2*a(n-4) + 10*a(n-5) + 4*a(n-6) - a(n-8) - a(n-9) for n > 9. (End)

Extensions

a(1)-a(2) and terms a(10) and beyond from Andrew Howroyd, Apr 16 2018

A302404 Total domination number of the n-Moebius ladder.

Original entry on oeis.org

0, 2, 2, 2, 3, 4, 4, 6, 6, 6, 7, 8, 8, 10, 10, 10, 11, 12, 12, 14, 14, 14, 15, 16, 16, 18, 18, 18, 19, 20, 20, 22, 22, 22, 23, 24, 24, 26, 26, 26, 27, 28, 28, 30, 30, 30, 31, 32, 32, 34, 34, 34, 35, 36, 36, 38, 38, 38, 39, 40, 40, 42, 42, 42, 43, 44, 44, 46, 46, 46, 47
Offset: 0

Views

Author

Eric W. Weisstein, Apr 07 2018

Keywords

Comments

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

Links

Crossrefs

Cf. A295420, A302405, A303046, A301337.

Programs

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

Formula

a(n) = (3 - (-1)^n + 4*n + cos(n*Pi/3) - 3*cos(2*n*Pi/3) + sqrt(3)*sin(n*Pi/3) + sin(2*n*Pi/3)/sqrt(3))/6.
a(n) = a(n-1) + a(n-6) - a(n-7).
G.f.: x*(2 + x^3 + x^4)/((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5)).
a(n) = a(n-6) + 4. - Andrew Howroyd, Apr 18 2018
a(n) = a(n-6*k) + 4*k. - Eric W. Weisstein, Apr 23 2018

A303046 Number of minimum total dominating sets in the n-Moebius ladder.

Original entry on oeis.org

1, 6, 9, 8, 25, 3, 196, 56, 9, 20, 121, 3, 1521, 154, 9, 32, 289, 3, 5776, 300, 9, 44, 529, 3, 15625, 494, 9, 56, 841, 3, 34596, 736, 9, 68, 1225, 3, 67081, 1026, 9, 80, 1681, 3, 118336, 1364, 9, 92, 2209, 3, 194481, 1750, 9, 104, 2809, 3, 302500, 2184, 9
Offset: 1

Views

Author

Eric W. Weisstein, Apr 17 2018

Keywords

Comments

Sequence extrapolated to n = 1 using recurrence. - Andrew Howroyd, Apr 18 2018

Links

Crossrefs

Cf. A295420, A302404.

Programs

  • Mathematica
    Table[Piecewise[{{3, Mod[n, 6] == 0}, {(n (n + 5)/6)^2, Mod[n, 6] == 1}, {n (2 n + 5)/3, Mod[n, 6] == 2}, {9, Mod[n, 6] == 3}, {2 n, Mod[n, 6] == 4}, {n^2, Mod[n, 6] == 5}}], {n, 200}]
LinearRecurrence[{0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, -10, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, -5, 0, 0, 0, 0, 0, 1}, {1, 6, 9, 8, 25, 3, 196, 56, 9, 20, 121, 3, 1521, 154, 9, 32, 289, 3, 5776, 300, 9, 44, 529, 3, 15625, 494, 9, 56, 841, 3}, 200]
Rest @ CoefficientList[Series[3 x^6/(1 - x^6) - 9 x^3/(-1 + x^6) + 4 x^4 (2 + x^6)/(-1 + x^6)^2 - x^5 (25 + 46 x^6 + x^12)/(-1 + x^6)^3 - 2 x^2 (3 + 19 x^6 + 2 x^12)/(-1 + x^6)^3 - x (1 + 191 x^6 + 551 x^12 + 121 x^18)/(-1 + x^6)^5, {x, 0, 200}], x]
  • PARI
    a(n)=my(k=n\6,r=n%6);if(r<3, if(r==0, 3, if(r==1, n^2*(k+1)^2, n*(4*k+3))), if(r==3, 9, if(r==4, 2*n, n^2))) \\ Andrew Howroyd, Apr 18 2018

Formula

From Andrew Howroyd, Apr 18 2018: (Start)
a(n) = 5*a(n-6) - 10*a(n-12) + 10*a(n-18) - 5*a(n-24) + a(n-30) for n > 30.
a(6k) = 3, a(6k+1) = (6*k+1)^2*(k+1)^2, a(6k+2) = (6*k+2)*(4*k+3), a(6k+3) = 9, a(6k+4) = (6*k+4)*2, a(6k+5) = (6*k+5)^2. (End)
a(3k) = 6 - 3*(-1)^k. - Eric W. Weisstein, Apr 19 2018

Extensions

a(1)-a(2) and terms a(14) and beyond from Andrew Howroyd, Apr 18 2018

A303162 Number of minimal total dominating sets in the n-Moebius ladder.

Original entry on oeis.org

0, 6, 9, 14, 25, 57, 196, 222, 441, 851, 1936, 3281, 6084, 12662, 24964, 48830, 93636, 188265, 369664, 725859, 1423249, 2798582, 5503716, 10790049, 21206025, 41601462, 81703521, 160396110, 314991504, 618413702, 1214104336, 2384319102, 4681706929, 9192838950
Offset: 1

Views

Author

Eric W. Weisstein, Apr 19 2018

Keywords

Comments

Sequence extrapolated to n=1 using recurrence.

Links

Crossrefs

Cf. A284663, A290337, A295420, A303006, A303046.

Programs

  • Mathematica
    Table[3 - 3 (-1)^n + RootSum[1 - 2 # - #^2 + 3 #^3 - #^4 - 2 #^5 + #^6 &, #^n &] - RootSum[1 - 4 # + 10 #^2 - 19 #^3 + 28 #^4 - 34 #^5 + 37 #^6 - 34 #^7 + 28 #^8 - 19 #^9 + 10 #^10 - 4 #^11 + #^12 &, #^n &] + RootSum[1 + 4 # + 10 #^2 + 19 #^3 + 28 #^4 + 34 #^5 + 37 #^6 + 34 #^7 + 28 #^8 + 19 #^9 + 10 #^10 + 4 #^11 + #^12 &, #^n &], {n, 200}]
LinearRecurrence[{2, -2, 3, 4, -7, 5, 0, -21, 39, -24, 21, 33, -36,
  63, -33, 0, 33, -63, 36, -33, -21, 24, -39, 21, 0, -5, 7, -4, -3,
  2, -2, 1}, {0, 6, 9, 14, 25, 57, 196, 222, 441, 851, 1936, 3281,
  6084, 12662, 24964, 48830, 93636, 188265, 369664, 725859, 1423249,
  2798582, 5503716, 10790049, 21206025, 41601462, 81703521, 160396110,314991504, 618413702, 1214104336, 2384319102}, 200]
Rest @ CoefficientList[Series[x^2 (6 - 3 x + 8 x^2 - 3 x^3 - 16 x^4 + 96 x^5 - 154 x^6 + 171 x^7 - 172 x^8 - 105 x^9 + 74 x^10 - 280 x^11 - 8 x^12 + 91 x^13 - 508 x^14 + 289 x^15 - 386 x^16 - 64 x^17 - 124 x^18 - 231 x^19 - 28 x^20 - 63 x^21 - 28 x^22 + 96 x^23 - 46 x^24 + 39 x^25 - 16 x^26 - 21 x^27 + 18 x^28 - 12 x^29 + 6 x^30)/((1 - x) (1 + x) (1 - 2 x - x^2 + 3 x^3 - x^4 - 2 x^5 + x^6) (1 - 4 x + 10 x^2 - 19 x^3 + 28 x^4 - 34 x^5 + 37 x^6 - 34 x^7 + 28 x^8 - 19 x^9 + 10 x^10 - 4 x^11 + x^12) (1 + 4 x + 10 x^2 + 19 x^3 + 28 x^4 + 34 x^5 + 37 x^6 + 34 x^7 + 28 x^8 + 19 x^9 + 10 x^10 + 4 x^11 + x^12)), {x, 0, 200}], x]

Formula

G.f.: x^2*(6 - 3*x + 8*x^2 - 3*x^3 - 16*x^4 + 96*x^5 - 154*x^6 + 171*x^7 - 172*x^8 - 105*x^9 + 74*x^10 - 280*x^11 - 8*x^12 + 91*x^13 - 508*x^14 + 289*x^15 - 386*x^16 - 64*x^17 - 124*x^18 - 231*x^19 - 28*x^20 - 63*x^21 - 28*x^22 + 96*x^23 - 46*x^24 + 39*x^25 - 16*x^26 - 21*x^27 + 18*x^28 - 12*x^29 + 6*x^30)/((1 - x)*(1 + x)*(1 - 2*x - x^2 + 3*x^3 - x^4 - 2*x^5 + x^6)*(1 - 4*x + 10*x^2 - 19*x^3 + 28*x^4 - 34*x^5 + 37*x^6 - 34*x^7 + 28*x^8 - 19*x^9 + 10*x^10 - 4*x^11 + x^12)*(1 + 4*x + 10*x^2 + 19*x^3 + 28*x^4 + 34*x^5 + 37*x^6 + 34*x^7 + 28*x^8 + 19*x^9 + 10*x^10 + 4*x^11 + x^12)). - Andrew Howroyd, Apr 19 2018

Extensions

a(1)-a(2) and terms a(11) and beyond from Andrew Howroyd, Apr 19 2018
Showing 1-4 of 4 results.

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