A198307 Moore lower bound on the order of a (7,g)-cage.

8, 14, 50, 86, 302, 518, 1814, 3110, 10886, 18662, 65318, 111974, 391910, 671846, 2351462, 4031078, 14108774, 24186470, 84652646, 145118822, 507915878, 870712934, 3047495270, 5224277606, 18284971622, 31345665638, 109709829734, 188073993830, 658258978406

Offset: 3

Author

Jason Kimberley, Oct 30 2011

Keywords

• nonn
• easy
• base

Links

• Colin Barker, Table of n, a(n) for n = 3..1000
• Gordon Royle, Cages of higher valency
• Index entries for linear recurrences with constant coefficients, signature (1,6,-6).

Crossrefs

Moore lower bound on the order of a (k,g) cage: A198300 (square); rows: A000027 (k=2), A027383 (k=3), A062318 (k=4), A061547 (k=5), A198306 (k=6), this sequence (k=7), A198308 (k=8), A198309 (k=9), A198310 (k=10), A094626 (k=11); columns: A020725 (g=3), A005843 (g=4), A002522 (g=5), A051890 (g=6), A188377 (g=7).

Programs

Mathematica

DeleteCases[CoefficientList[Series[2 x^3*(4 + 3 x - 6 x^2)/((1 - x) (1 - 6 x^2)), {x, 0, 31}], x], 0] (* Michael De Vlieger, Mar 17 2017 *)
LinearRecurrence[{1,6,-6},{8,14,50},30] (* or *) CoefficientList[ Series[ -((2 (-4-3 x+6 x^2))/(1-x-6 x^2+6 x^3)),{x,0,30}],x] (* Harvey P. Dale, Aug 03 2021 *)

PARI

Vec(2*x^3*(4 + 3*x - 6*x^2) / ((1 - x)*(1 - 6*x^2)) + O(x^40)) \\ Colin Barker, Mar 17 2017

Formula

a(2*i) = 2*Sum_{j=0..i-1}6^j = string "2"^i read in base 6.

a(2*i+1) = 6^i + 2*Sum_{j=0..i-1}6^j = string "1"*"2"^i read in base 6.

a(n) <= A218555(n).

From Colin Barker, Feb 01 2013: (Start)

a(n) = a(n-1) + 6*a(n-2) - 6*a(n-3) for n>5.

G.f.: 2*x^3*(4 + 3*x - 6*x^2) / ((1 - x)*(1 - 6*x^2)). (End)

From Colin Barker, Mar 17 2017: (Start)

a(n) = 2*(6^(n/2) - 1)/5 for n>2 and even.

a(n) = (7*6^(n/2-1/2) - 2)/5 for n>2 and odd. (End)

E.g.f.: (12*(cosh(sqrt(6)*x) - cosh(x)) + 7*sqrt(6)*sinh(sqrt(6)*x) - 12*sinh(x) - 30*x*(1 + x))/30. - Stefano Spezia, Apr 07 2022