A198309 Moore lower bound on the order of a (9,g)-cage.

10, 18, 82, 146, 658, 1170, 5266, 9362, 42130, 74898, 337042, 599186, 2696338, 4793490, 21570706, 38347922, 172565650, 306783378, 1380525202, 2454267026, 11044201618, 19634136210, 88353612946, 157073089682, 706828903570, 1256584717458, 5654631228562

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 (1,8,-8).

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), A198308 (k=8), this sequence (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[{1,8,-8},{10,18,82},30] (* Harvey P. Dale, Apr 03 2015 *)

PARI

Vec(2*x^3*(5 + 4*x - 8*x^2) / ((1 - x)*(1 - 8*x^2)) + O(x^40)) \\ Colin Barker, Mar 17 2017

Formula

a(2*i) = 2 Sum_{j=0..i-1} 8^j = string "2"^i read in base 8.

a(2*i+1) = 8^i + 2 Sum_{j=0..i-1} 8^j = string "1"*"2"^i read in base 8.

From Colin Barker, Feb 01 2013: (Start)

a(n) = a(n-1) + 8*a(n-2) - 8*a(n-3) for n>5.

G.f.: 2*x^3*(5 + 4*x - 8*x^2) / ((1 - x)*(1 - 8*x^2)). (End)

From Colin Barker, Mar 17 2017: (Start)

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

a(n) = (9*2^((3*(n-1))/2) - 2)/7 for n odd. (End)

E.g.f.: (8*(cosh(2*sqrt(2)*x) - cosh(x) - sinh(x)) + 9*sqrt(2)*sinh(2*sqrt(2)*x) - 28*x*(1 + x))/28. - Stefano Spezia, Apr 09 2022