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%I #11 Dec 31 2022 13:39:59
%S 1,7,43,253,1465,8431,48403,277621,1591729,9124759,52305595,299822893,
%T 1718610409,9851185663,56467549987,323674986277,1855321740385,
%U 10634799101479,60959210186827,349421293175389,2002900612974361,11480728092514831,65808116771451955,377215468968922837
%N a(n) = ((3+sqrt(3))*(4+sqrt(3))^n-(3-sqrt(3))*(4-sqrt(3))^n)/sqrt(12).
%C Fourth binomial transform of A162436, inverse binomial transform of A162272.
%C The inverse binomial transform yields A030192. The binomial transform yields A162272. - _R. J. Mathar_, Jul 07 2009
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8,-13).
%F a(n) = 8*a(n-1) - 13(n-2) for n > 1; a(0) = 1, a(1) = 7.
%F G.f.: (1 - x)/(1 - 8*x + 13*x^2). - _Klaus Brockhaus_, Jun 19 2009
%F E.g.f.: exp(4*x)*(cosh(sqrt(3)*x) + sqrt(3)*sinh(sqrt(3)*x)). - _Stefano Spezia_, Dec 31 2022
%o (PARI) F=nfinit(x^2-3); for(n=0, 20, print1(nfeltdiv(F, ((3+x)*(4+x)^n-(3-x)*(4-x)^n), (2*x))[1], ",")) \\ _Klaus Brockhaus_, Jun 19 2009
%o (PARI) Vec((1-x)/(1-8*x+13*x^2)+O(x^25)) \\ _M. F. Hasler_, Dec 03 2014
%Y Cf. A162436, A162272, A030192, A161729.
%K nonn,easy
%O 0,2
%A Al Hakanson (hawkuu(AT)gmail.com), Jun 17 2009
%E Extended beyond a(5) by _Klaus Brockhaus_, Jun 19 2009
%E Edited by _Klaus Brockhaus_, Jul 05 2009; _M. F. Hasler_, Dec 03 2014
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