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%I #34 Apr 10 2022 15:35:51
%S 8,14,50,86,302,518,1814,3110,10886,18662,65318,111974,391910,671846,
%T 2351462,4031078,14108774,24186470,84652646,145118822,507915878,
%U 870712934,3047495270,5224277606,18284971622,31345665638,109709829734,188073993830,658258978406
%N Moore lower bound on the order of a (7,g)-cage.
%H Colin Barker, <a href="/A198307/b198307.txt">Table of n, a(n) for n = 3..1000</a>
%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 (1,6,-6).
%F a(2*i) = 2*Sum_{j=0..i-1}6^j = string "2"^i read in base 6.
%F a(2*i+1) = 6^i + 2*Sum_{j=0..i-1}6^j = string "1"*"2"^i read in base 6.
%F a(n) <= A218555(n).
%F From _Colin Barker_, Feb 01 2013: (Start)
%F a(n) = a(n-1) + 6*a(n-2) - 6*a(n-3) for n>5.
%F G.f.: 2*x^3*(4 + 3*x - 6*x^2) / ((1 - x)*(1 - 6*x^2)). (End)
%F From _Colin Barker_, Mar 17 2017: (Start)
%F a(n) = 2*(6^(n/2) - 1)/5 for n>2 and even.
%F a(n) = (7*6^(n/2-1/2) - 2)/5 for n>2 and odd. (End)
%F 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
%t 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 *)
%t 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 *)
%o (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
%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), 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).
%K nonn,easy,base
%O 3,1
%A _Jason Kimberley_, Oct 30 2011
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