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

%I A198309 #25 Apr 10 2022 15:35:58
%S A198309 10,18,82,146,658,1170,5266,9362,42130,74898,337042,599186,2696338,
%T A198309 4793490,21570706,38347922,172565650,306783378,1380525202,2454267026,
%U A198309 11044201618,19634136210,88353612946,157073089682,706828903570,1256584717458,5654631228562
%N A198309 Moore lower bound on the order of a (9,g)-cage.
%H A198309 Colin Barker, <a href="/A198309/b198309.txt">Table of n, a(n) for n = 3..1000</a>
%H A198309 Gordon Royle, <a href="http://staffhome.ecm.uwa.edu.au/~00013890/remote/cages/allcages.html">Cages of higher valency</a>
%H A198309 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,8,-8).
%F A198309 a(2*i) = 2 Sum_{j=0..i-1} 8^j =  string "2"^i read in base 8.
%F A198309 a(2*i+1) = 8^i + 2 Sum_{j=0..i-1} 8^j = string "1"*"2"^i read in base 8.
%F A198309 From _Colin Barker_, Feb 01 2013: (Start)
%F A198309 a(n) = a(n-1) + 8*a(n-2) - 8*a(n-3) for n>5.
%F A198309 G.f.: 2*x^3*(5 + 4*x - 8*x^2) / ((1 - x)*(1 - 8*x^2)). (End)
%F A198309 From _Colin Barker_, Mar 17 2017: (Start)
%F A198309 a(n) = 2*(2^(3*n/2) - 1)/7 for n even.
%F A198309 a(n) = (9*2^((3*(n-1))/2) - 2)/7 for n odd. (End)
%F A198309 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
%t A198309 LinearRecurrence[{1,8,-8},{10,18,82},30] (* _Harvey P. Dale_, Apr 03 2015 *)
%o A198309 (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
%Y A198309 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).
%K A198309 nonn,easy,base
%O A198309 3,1
%A A198309 _Jason Kimberley_, Oct 30 2011