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.

A290337 Number of minimal dominating sets in the n-Moebius ladder.

Original entry on oeis.org

2, 4, 11, 28, 37, 67, 149, 284, 596, 1179, 2444, 5023, 10103, 20577, 41746, 84860, 172501, 350392, 712597, 1448463, 2943959, 5983960, 12162310, 24721031, 50246512, 102128407, 207584129, 421927877, 857596064, 1743119352, 3543000201, 7201377180, 14637255611
Offset: 1

Views

Author

Eric W. Weisstein, Jul 27 2017

Keywords

Links

Crossrefs

Cf. A284663, A290336, A290379.

Formula

Empirical: a(n) = a(n-1)+a(n-2)+2*a(n-3) -a(n-4)+2*a(n-5)-2*a(n-6) +6*a(n-7)+4*a(n-8)+4*a(n-9) -6*a(n-10)-3*a(n-12) +5*a(n-13)-a(n-14)-2*a(n-15) -5*a(n-16)-2*a(n-17)-2*a(n-18) for n > 18. - Andrew Howroyd, Aug 01 2017
Empirical g.f.: x*(2 + 2*x + 5*x^2 + 9*x^3 - 8*x^4 - 20*x^5 - 4*x^6 - 4*x^7 - 40*x^9 - 26*x^10 - 26*x^11 + 14*x^12 - 22*x^13 - 33*x^14 - 45*x^15 - 14*x^16 - 14*x^17) / ((1 - x)*(1 + 2*x^4 + x^6)*(1 - x^2 - 3*x^3 - 4*x^4 - 4*x^5 - x^6 - 2*x^7 - 3*x^8 - 5*x^9 - 4*x^10 - 2*x^11)). - Colin Barker, Aug 02 2017

Extensions

a(1)-a(2) and terms a(9) and beyond from Andrew Howroyd, Aug 01 2017

A290379 Number of minimal dominating sets in the n-ladder graph.

Original entry on oeis.org

2, 6, 7, 18, 39, 75, 155, 310, 638, 1295, 2624, 5339, 10853, 22069, 44836, 91134, 185259, 376542, 765331, 1555567, 3161843, 6426646, 13062506, 26550391, 53965428, 109688223, 222948193, 453156469, 921069708, 1872133138, 3805230243, 7734373962, 15720610559
Offset: 1

Views

Author

Eric W. Weisstein, Jul 28 2017

Keywords

Links

Crossrefs

Row 2 of A286847.
Cf. A290336, A290337.

Programs

  • Magma
    I:=[2,6,7,18,39,75,155,310,638,1295,2624]; [n le 11 select I[n] else Self(n-2)+3*Self(n-3)+4*Self(n-4)+4*Self(n-5)+Self(n-6)+2*Self(n-7)+3*Self(n-8)+5*Self(n-9)+4*Self(n-10)+2*Self(n-11): n in [1..40]]; // Vincenzo Librandi, Aug 04 2017
  • Mathematica
    Table[-RootSum[-2 - 4 # - 5 #^2 - 3 #^3 - 2 #^4 - #^5 - 4 #^6 - 4 #^7 - 3 #^8 - #^9 + #^11 &, 621827501801 #^n - 301456826961 #^(n + 1) + 280366986955 #^(n + 2) - 1253389979482 #^(n + 3) + 843186094854 #^(n + 4) - 87555893434 #^(n + 5) + 236346312907 #^(n + 6) - 504072574383 #^(n + 7) + 231943645265 #^(n + 8) - 618185916584 #^(n + 9) + 290649224768 #^(n + 10) &]/2097121971853, {n, 20}] (* Eric W. Weisstein, Aug 04 2017 *)
LinearRecurrence[{0, 1, 3, 4, 4, 1, 2, 3, 5, 4, 2}, {2, 6, 7, 18, 39, 75, 155, 310, 638, 1295, 2624}, 20] (* Eric W. Weisstein, Aug 04 2017 *)
CoefficientList[Series[((1 + x) (2 + 4 x + x^2 + 5 x^3 + x^4 + 3 x^5 + 5 x^6 + 3 x^7 + 2 x^8 + 2 x^9))/(1 - x^2 - 3 x^3 - 4 x^4 - 4 x^5 - x^6 - 2 x^7 - 3 x^8 - 5 x^9 - 4 x^10 - 2 x^11), {x, 0, 20}], x] (* Eric W. Weisstein, Aug 04 2017 *)
  • PARI
    Vec((1+x)*(2+4*x+x^2+5*x^3+x^4+3*x^5+5*x^6+3*x^7+2*x^8+2*x^9)/(1-x^2-3*x^3-4*x^4-4*x^5-x^6-2*x^7-3*x^8-5*x^9-4*x^10-2*x^11)+O(x^40)) \\ Andrew Howroyd, Aug 01 2017

Formula

From Andrew Howroyd, Aug 01 2017: (Start)
a(n) = a(n-2) + 3*a(n-3) + 4*a(n-4) + 4*a(n-5) + a(n-6) + 2*a(n-7) + 3*a(n-8) + 5*a(n-9) + 4*a(n-10) + 2*a(n-11) for n > 11.
G.f.: x*(1+x)*(2 + 4*x + x^2 + 5*x^3 + x^4 + 3*x^5 + 5*x^6 + 3*x^7 + 2*x^8 + 2*x^9)/(1 - x^2 - 3*x^3 - 4*x^4 - 4*x^5 - x^6 - 2*x^7- 3*x^8 - 5*x^9 - 4*x^10 - 2*x^11).
(End)

Extensions

Terms a(9) and beyond from Andrew Howroyd, Aug 01 2017

A290377 Number of minimal dominating sets in the n-antiprism graph.

Original entry on oeis.org

4, 15, 12, 25, 55, 112, 188, 438, 789, 1573, 3135, 5980, 11848, 23035, 45020, 87873, 171910, 335464, 655397, 1281190, 2501173, 4888098, 9548543, 18653025, 36441500, 71190933, 139076320, 271694910, 530784135, 1036914040, 2025703900, 3957367099, 7731003525
Offset: 2

Views

Author

Eric W. Weisstein, Jul 28 2017

Keywords

Links

Crossrefs

Cf. A284699, A290336.

Formula

Empirical: a(n) = 2*a(n-2)+5*a(n-3)+a(n-4) -5*a(n-5)-8*a(n-6)+a(n-7) +6*a(n-8)+10*a(n-9)-2*a(n-10) -2*a(n-11)-5*a(n-12)+a(n-15) for n>16. - Andrew Howroyd, Aug 01 2017
Empirical g.f.: x^2*(4 + 15*x + 4*x^2 - 25*x^3 - 48*x^4 + 7*x^5 + 48*x^6 + 90*x^7 - 20*x^8 - 22*x^9 - 60*x^10 + 15*x^13) / (1 - 2*x^2 - 5*x^3 - x^4 + 5*x^5 + 8*x^6 - x^7 - 6*x^8 - 10*x^9 + 2*x^10 + 2*x^11 + 5*x^12 - x^15). - Colin Barker, Aug 01 2017

Extensions

a(2) and terms a(8) and beyond from Andrew Howroyd, Aug 01 2017

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

A303006 Number of minimal total dominating sets in the n-prism graph.

Original entry on oeis.org

2, 4, 5, 36, 27, 25, 114, 196, 437, 729, 1674, 3249, 6450, 12996, 24870, 49284, 95882, 190969, 369666, 724201, 1425261, 2802276, 5495162, 10764961, 21186827, 41602500, 81686669, 160326244, 314946266, 618516900, 1214288106, 2384368900, 4681737021, 9193357924
Offset: 1

Views

Author

Eric W. Weisstein, Apr 17 2018

Keywords

Comments

Sequence extrapolated to n=1 using recurrence. - Andrew Howroyd, Apr 17 2018

Links

Crossrefs

Cf. A290336, A296102, A302405, A303053.

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}, {2, 4, 5, 36, 27, 25, 114, 196, 437, 729, 1674, 3249,
  6450, 12996, 24870, 49284, 95882, 190969, 369666, 724201, 1425261,
  2802276, 5495162, 10764961, 21186827, 41602500, 81686669, 160326244, 314946266, 618516900, 1214288106, 2384368900}, 200]
CoefficientList[Series[(2 + x^2 + 28 x^3 - 55 x^4 + 26 x^5 + 8 x^6 - 192 x^7 + 359 x^8 - 180 x^9 + 83 x^10 + 552 x^11 - 700 x^12 + 906 x^13 - 583 x^14 - 228 x^15 + 605 x^16 - 1362 x^17 + 596 x^18 - 636 x^19 - 673 x^20 + 684 x^21 - 1045 x^22 + 564 x^23 + 8 x^24 - 154 x^25 + 197 x^26 - 116 x^27 - 107 x^28 + 72 x^29 - 70 x^30 + 36 x^31)/((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, 199}], x]

Formula

G.f.: x*(2 + x^2 + 28*x^3 - 55*x^4 + 26*x^5 + 8*x^6 - 192*x^7 + 359*x^8 - 180*x^9 + 83*x^10 + 552*x^11 - 700*x^12 + 906*x^13 - 583*x^14 - 228*x^15 + 605*x^16 - 1362*x^17 + 596*x^18 - 636*x^19 - 673*x^20 + 684*x^21 - 1045*x^22 + 564*x^23 + 8*x^24 - 154*x^25 + 197*x^26 - 116*x^27 - 107*x^28 + 72*x^29 - 70*x^30 + 36*x^31)/((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 17 2018

Extensions

a(1)-a(2) and terms a(10) and beyond from Andrew Howroyd, Apr 17 2018

A290381 Number of minimal dominating sets in the n-web graph.

Original entry on oeis.org

22, 53, 146, 338, 995, 2661, 6961, 18770, 50161, 134426, 359126, 960419, 2570837, 6875493, 18392182, 49200125, 131613970, 352077098, 941809667, 2519398997, 6739522745, 18028532346, 48227208121, 129010104410, 345108392014, 923181669827, 2469555755813
Offset: 3

Views

Author

Eric W. Weisstein, Jul 28 2017

Keywords

Links

Crossrefs

Cf. A287474, A290336.

Formula

Empirical: a(n) = a(n-1)+2*a(n-2)+5*a(n-3) +4*a(n-4)+4*a(n-5)-8*a(n-6) for n>8. - Andrew Howroyd, Aug 01 2017
Empirical g.f.: x^3*(22 + 31*x + 49*x^2 - 24*x^3 + 12*x^4 - 40*x^5) / (1 - x - 2*x^2 - 5*x^3 - 4*x^4 - 4*x^5 + 8*x^6). - Colin Barker, Aug 01 2017

Extensions

Terms a(7) and beyond from Andrew Howroyd, Aug 01 2017

A290511 Number of irredundant sets in the n-prism graph.

Original entry on oeis.org

3, 9, 24, 77, 198, 522, 1550, 4477, 12732, 36214, 103579, 296294, 846303, 2417368, 6907329, 19737901, 56396602, 161138214, 460420182, 1315565162, 3758963099, 10740463083, 30688703123, 87686813998, 250547405698, 715888629491, 2045507376543, 5844625043236
Offset: 1

Views

Author

Eric W. Weisstein, Aug 04 2017

Keywords

Comments

The n-prism graph is well defined for n >= 3. Sequence extended to n=1 via the number of period n periodic solutions on a larger graph. - Andrew Howroyd, Aug 07 2017

Links

Crossrefs

Cf. A284702, A290336, A290493.

Formula

Empirical: a(n) = 3*a(n-1) - a(n-3) + 2*a(n-4) - 6*a(n-5) - 8*a(n-6) + 13*a(n-7) + 6*a(n-8) - 6*a(n-9) - 2*a(n-10) - 5*a(n-11) - 4*a(n-12) + 2*a(n-13) + 2*a(n-14) - a(n-15) + a(n-16) + 3*a(n-17) + 2*a(n-19) for n > 19. - Andrew Howroyd, Aug 07 2017

Extensions

a(1)-a(2) and terms a(10) and beyond from Andrew Howroyd, Aug 07 2017
Showing 1-7 of 7 results.

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