A198306 Moore lower bound on the order of a (6,g)-cage.

7, 12, 37, 62, 187, 312, 937, 1562, 4687, 7812, 23437, 39062, 117187, 195312, 585937, 976562, 2929687, 4882812, 14648437, 24414062, 73242187, 122070312, 366210937, 610351562, 1831054687, 3051757812, 9155273437, 15258789062

Offset: 3

Author

Jason Kimberley, Oct 30 2011

Keywords

• nonn
• easy
• base

Links

• Gordon Royle, Cages of higher valency
• Index entries for linear recurrences with constant coefficients, signature (1,5,-5).

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), this sequence (k=6), A198307 (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

LinearRecurrence[{1,5,-5},{7,12,37},30] (* Harvey P. Dale, Jun 28 2015 *)

Formula

a(2*i) = 2*Sum_{j=0..i-1} 5^j = string "2"^i read in base 5.

a(2*i+1) = 5^i + 2*Sum_{j=0..i-1} 5^j = string "1"*"2"^i read in base 5.

a(n) <= A218554(n). - Jason Kimberley, Dec 21 2012

a(n) = a(n-1)+5*a(n-2)-5*a(n-3). G.f.: -x^3*(10*x^2-5*x-7) / ((x-1)*(5*x^2-1)). - Colin Barker, Feb 01 2013

From Colin Barker, Nov 25 2016: (Start)

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

a(n) = (3*5^((n-1)/2) - 1)/2 for n>2 and odd. (End)

E.g.f.: (5*cosh(sqrt(5)*x) - 5*cosh(x) - 5*sinh(x) + 3*sqrt(5)*sinh(sqrt(5)*x) - 10*x*(1 + x))/10. - Stefano Spezia, Apr 07 2022