A198308 Moore lower bound on the order of an (8,g)-cage.

9, 16, 65, 114, 457, 800, 3201, 5602, 22409, 39216, 156865, 274514, 1098057, 1921600, 7686401, 13451202, 53804809, 94158416, 376633665, 659108914, 2636435657, 4613762400, 18455049601, 32296336802, 129185347209, 226074357616, 904297430465, 1582520503314

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 (1,7,-7).

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), A198307 (k=7), this sequence (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

LinearRecurrence[{1,7,-7},{9,16,65},40] (* Harvey P. Dale, Oct 14 2019 *)

PARI

Vec(x^3*(9 + 7*x - 14*x^2) / ((1 - x)*(1 - 7*x^2)) + O(x^40)) \\ Colin Barker, Mar 17 2017

Formula

a(2*i) = 2 Sum_{j=0..i-1} 7^j = string "2"^i read in base 7.

a(2*i+1) = 7^i + 2 Sum_{j=0..i-1} 7^j = string "1"*"2"^i read in base 7.

From Colin Barker, Feb 01 2013: (Start)

a(n) = a(n-1) + 7*a(n-2) - 7*a(n-3) for n>5.

G.f.: x^3*(9 + 7*x - 14*x^2) / ((1 - x)*(1 - 7*x^2)). (End)

From Colin Barker, Mar 17 2017: (Start)

a(n) = (7^(n/2) - 1)/3 for n even.

a(n) = (4*7^(n/2-1/2) - 1)/3 for n odd. (End)

E.g.f.: (7*(cosh(sqrt(7)*x) - cosh(x) - sinh(x)) + 4*sqrt(7)*sinh(sqrt(7)*x) - 21*x*(1 + x))/21. - Stefano Spezia, Apr 09 2022