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%I #13 Jan 04 2022 21:01:05
%S 289,4225,139129,4721929,160402225,5448949489,185103876169,
%T 6288082836025,213609712544449,7256442143671009,246505423172265625,
%U 8373927945713356009,284467044731081834449,9663505592911069011025
%N a(n) = 34*a(n-1)-a(n-2)-4232 for n > 2; a(1)=289, a(2)=4225.
%C lim_{n -> infinity} a(n)/a(n-1) = 17+12*sqrt(2).
%H G. C. Greubel, <a href="/A156575/b156575.txt">Table of n, a(n) for n = 1..600</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (35,-35,1).
%F a(n) = (1058 + (4947 - 3478*sqrt(2))*(17 + 12*sqrt(2))^n + (4947 + 3478*sqrt(2))*(17 - 12*sqrt(2))^n)/8.
%F G.f.: x*(289 -5890*x +1369*x^2)/((1-x)*(1-34*x+x^2)).
%F a(1)=289, a(2)=4225, a(3)=139129, a(n) = 35*a(n-1) - 35*a(n-2) + a(n-3). - _Harvey P. Dale_, Dec 15 2011
%F a(n) = -1369*[n=0] + (529/4) + (3/4)*(1649*ChebyshevU(n, 17) - 55857*ChebyshevU(n-1, 17)). - _G. C. Greubel_, Jan 04 2022
%e a(4) = 34*a(3) -a(2) -4232 = 34*139129 -4225 -4232 = 4721929.
%t RecurrenceTable[{a[1]==289,a[2]==4225,a[n]==34a[n-1]-a[n-2]-4232},a,{n,20}] (* or *) LinearRecurrence[{35,-35,1},{289,4225,139129},20] (* _Harvey P. Dale_, Dec 15 2011 *)
%o (PARI) {m=14; v=concat([289, 4225], vector(m-2)); for(n=3, m, v[n]=34*v[n-1]-v[n-2]-4232); v}
%o (Sage)
%o def a(n): return -1369*bool(n==0) + (529/4) + (3/4)*(1649*chebyshev_U(n, 17) - 55857*chebyshev_U(n-1, 17))
%o [a(n) for n in (1..30)] # _G. C. Greubel_, Jan 04 2022
%Y First trisection of A156572.
%Y Cf. A156164 (decimal expansion of 17+12*sqrt(2)), A156573, A156574.
%K nonn,easy
%O 1,1
%A _Klaus Brockhaus_, Feb 11 2009
%E Revised by _Klaus Brockhaus_, Feb 16 2009
%E G.f. corrected by _Klaus Brockhaus_, Sep 22 2009
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