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

2,1

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

Table of n, a(n) for n=2..31.

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).

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

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}]

Crossrefs

Cf. A362545.

Sequence in context: A196538 A216418 A254626 * A262769 A298393 A299511

Adjacent sequences: A362537 A362538 A362539 * A362541 A362542 A362543

Keyword

nonn,easy

Author

Eric W. Weisstein, Apr 24 2023

Extensions

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

Status

approved