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%I #35 Apr 10 2022 15:35:43
%S 7,12,37,62,187,312,937,1562,4687,7812,23437,39062,117187,195312,
%T 585937,976562,2929687,4882812,14648437,24414062,73242187,122070312,
%U 366210937,610351562,1831054687,3051757812,9155273437,15258789062
%N Moore lower bound on the order of a (6,g)-cage.
%H Gordon Royle, <a href="http://staffhome.ecm.uwa.edu.au/~00013890/remote/cages/allcages.html">Cages of higher valency</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,5,-5).
%F a(2*i) = 2*Sum_{j=0..i-1} 5^j = string "2"^i read in base 5.
%F a(2*i+1) = 5^i + 2*Sum_{j=0..i-1} 5^j = string "1"*"2"^i read in base 5.
%F a(n) <= A218554(n). - _Jason Kimberley_, Dec 21 2012
%F 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
%F From _Colin Barker_, Nov 25 2016: (Start)
%F a(n) = (5^(n/2) - 1)/2 for n>2 and even.
%F a(n) = (3*5^((n-1)/2) - 1)/2 for n>2 and odd. (End)
%F 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
%t LinearRecurrence[{1,5,-5},{7,12,37},30] (* _Harvey P. Dale_, Jun 28 2015 *)
%Y 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).
%K nonn,easy,base
%O 3,1
%A _Jason Kimberley_, Oct 30 2011
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