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 1-6 of 6 results.

A290336 Number of minimal dominating sets in the n-prism graph.

Original entry on oeis.org

11, 12, 37, 55, 149, 316, 596, 1219, 2444, 4971, 10103, 20465, 41746, 84924, 172501, 350668, 712597, 1448447, 2943959, 5983344, 12162310, 24720787, 50246512, 102129655, 207584129, 421928981, 857596064, 1743117100, 3543000201, 7201373724, 14637255611
Offset: 3

Views

Author

Eric W. Weisstein, Jul 27 2017

Keywords

Comments

The prism graphs are defined for n>=3. If the sequence is extended to n=1 using P_n X P_2 then a(1)=2 and a(2)=6 (as A290379). The empirical recurrence is the same as that for the Moebius ladder graph (see A290337). - Andrew Howroyd, Aug 01 2017

Links

Crossrefs

Cf. A284702, A290337, 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 > 20. - Andrew Howroyd, Aug 01 2017
Empirical g.f.: x^3*(11 + x + 14*x^2 - 16*x^3 + 44*x^4 + 28*x^5 + 56*x^6 - 52*x^7 - 6*x^8 - 70*x^9 + 52*x^10 - 28*x^11 - 23*x^12 - 97*x^13 - 56*x^14 - 62*x^15 - 12*x^16 - 8*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

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

A295420 Number of total dominating sets in the n-Moebius ladder.

Original entry on oeis.org

1, 11, 49, 131, 441, 1499, 5041, 17155, 58081, 196331, 664225, 2246915, 7601049, 25714875, 86992929, 294294531, 995591809, 3368061131, 11394068049, 38545859971, 130399709881, 441139059867, 1492362754129, 5048627019523, 17079382863841, 57779138374059
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. A284663, A290337.

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}, {1, 11, 49, 131, 441, 1499, 5041, 17155, 58081}, 20]
CoefficientList[Series[(1 + 8 x + 16 x^2 - 20 x^3 + 6 x^4 - 8 x^5 + 4 x^6 - 4 x^7 - 3 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((1 - x)*(1 + 9*x + 25*x^2 + 5*x^3 + 11*x^4 + 3*x^5 + 7*x^6 + 3*x^7)/((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*(1 - x)*(1 + 9*x + 25*x^2 + 5*x^3 + 11*x^4 + 3*x^5 + 7*x^6 + 3*x^7)/((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

A290509 Number of irredundant sets in the n-Moebius ladder.

Original entry on oeis.org

3, 5, 24, 81, 198, 542, 1550, 4497, 12732, 36210, 103579, 296250, 846303, 2417308, 6907329, 19737889, 56396602, 161138306, 460420182, 1315565326, 3758963099, 10740463167, 30688703123, 87686813826, 250547405698, 715888629071, 2045507376543, 5844625042904
Offset: 1

Views

Author

Eric W. Weisstein, Aug 04 2017

Keywords

Links

Crossrefs

Cf. A284663, A290337.

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 05 2017
Empirical g.f.: x*(3 - 4*x + 9*x^2 + 12*x^3 - 46*x^4 - 20*x^5 + 11*x^6 + 28*x^7 - 18*x^8 - 8*x^9 + 21*x^10 - 4*x^11 + 18*x^12 + 8*x^13 - 15*x^14 + 4*x^15 + 23*x^16 + 14*x^18) / ((1 - x)*(1 - x^2 + x^4 + x^6)*(1 - 2*x - x^2 - 3*x^3 - 5*x^4 + 2*x^5 + 6*x^6 + 5*x^7 + 4*x^8 + 4*x^9 + 3*x^10 + 2*x^11 + 2*x^12)). - Colin Barker, Aug 05 2017

Extensions

a(1)-a(2) and terms a(10) and beyond from Andrew Howroyd, Aug 05 2017

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

A347559 Number of minimum dominating sets in the n-Moebius ladder.

Original entry on oeis.org

9, 24, 10, 4, 14, 80, 18, 4, 22, 168, 26, 4, 30, 288, 34, 4, 38, 440, 42, 4, 46, 624, 50, 4, 54, 840, 58, 4, 62, 1088, 66, 4, 70, 1368, 74, 4, 78, 1680, 82, 4, 86, 2024, 90, 4, 94, 2400, 98, 4, 102, 2808, 106, 4, 110, 3248, 114, 4, 118, 3720, 122, 4, 126, 4224
Offset: 3

Views

Author

Eric W. Weisstein, Sep 06 2021

Keywords

Links

Crossrefs

Cf. A284663, A290337.

Programs

  • Mathematica
    Table[Piecewise[{{9, n == 3}, {n (n + 2), Mod[n, 4] == 0}, {2 n, Mod[n, 2] == 1}, {4, Mod[n, 4] == 2}}, 0], {n, 3, 20}]
Join[{9}, LinearRecurrence[{0, 0, 0, 3, 0, 0, 0, -3, 0, 0, 0, 1}, {24, 10, 4, 14, 80, 18, 4, 22, 168, 26, 4, 30}, 20]]
CoefficientList[Series[(-9 - 24 x - 10 x^2 - 4 x^3 + 13 x^4 - 8 x^5 + 12 x^6 + 8 x^7 - 7 x^8 - 2 x^10 - 4 x^11 + 3 x^12)/((-1 + x)^3 (1 + x)^3 (1 + x^2)^3), {x, 0, 20}], x]

Formula

a(n) = n*(n+2) for n == 0 (mod 4).
a(n) = 2*n for n == 1 (mod 2) and n > 3.
a(n) = 4 for n == 2 (mod 4).
a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12) for n > 3.
G.f.: x^3*(-9 - 24*x - 10*x^2 - 4*x^3 + 13*x^4 - 8*x^5 + 12*x^6 + 8*x^7 - 7*x^8 - 2*x^10 - 4*x^11 + 3*x^12)/((-1 + x)^3*(1 + x)^3*(1 + x^2)^3).
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.