A362540 Number of chordless cycles of length >= 4 in the n-flower graph.

3, 23, 63, 127, 273, 583, 1287, 2975, 6993, 16535, 39525, 95071, 229029, 552199, 1332375, 3215807, 7762611, 18739607, 45240309, 109217983, 263673699, 636563527, 1536798717, 3710157407, 8957109801, 21624374039, 52205854257, 126036078751, 304278008331, 734592089095

Offset: 2

Author

Eric W. Weisstein, Apr 24 2023

Keywords

• nonn
• easy

Comments

The n-flower graph can be defined for n >= 3 without multiple edges. It is a snark for odd n >= 5. The sequence has been extended to n=2 using the recurrence. - Andrew Howroyd, Apr 26 2023

Links

• Eric Weisstein's World of Mathematics, Chordless Cycle
• Eric Weisstein's World of Mathematics, Flower Graph
• Index entries for linear recurrences with constant coefficients, signature (4,-5,4,-2,-4,7,-8,6,4,-6,0,1).

Crossrefs

Cf. A362545.

Programs

Mathematica

LinearRecurrence[{4, -5, 4, -2, -4, 7, -8, 6, 4, -6, 0, 1}, {3, 23, 63, 127, 273, 583, 1287, 2975, 6993, 16535, 39525, 95071}, 20]
CoefficientList[Series[(-3 - 11 x + 14 x^2 + 22 x^3 + 6 x^4 + 68 x^5 - 9 x^6 - 19 x^7 + 25 x^8 - 13 x^9 - 9 x^10 + x^11)/((-1 + x)^3 (1 - x - x^2 - 3 x^3 - 5 x^4 - 3 x^5 - 4 x^6 + 3 x^8 + x^9)), {x, 0, 20}], x]
Table[(1 + (-1)^n)/2 + 2 (-I)^n ChebyshevT[n, I] + 3 (n - 2) n + RootSum[-1 + # + #^3 &, #^n &] + RootSum[1 + # + #^3 &, #^n &], {n, 2, 20}]

Formula

From Andrew Howroyd, Apr 26 2023: (Start)

a(n) = 4*a(n-1) - 5*a(n-2) + 4*a(n-3) - 2*a(n-4) - 4*a(n-5) + 7*a(n-6) - 8*a(n-7) + 6*a(n-8) + 4*a(n-9) - 6*a(n-10) + a(n-12).

G.f.: x^2*(3 + 11*x - 14*x^2 - 22*x^3 - 6*x^4 - 68*x^5 + 9*x^6 + 19*x^7 - 25*x^8 + 13*x^9 + 9*x^10 - x^11)/((1 - x)^3*(1 + x)*(1 - 2*x - x^2)*(1 + x^2 - x^3)*(1 + x^2 + x^3)). (End)

2*a(n) = -2*(-1)^n*A112455(n) +1+6*n^2-12*n+(-1)^n-2*A112455(n)+4*A001333(n). - R. J. Mathar, Feb 18 2024

Extensions

a(2)-a(4) and a(17) and beyond from Andrew Howroyd, Apr 26 2023