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A156573 a(n) = 34*a(n-1) - a(n-2) - 4232 for n > 2; a(1)=529, a(2)=13225.

%I A156573 #13 Jan 07 2022 07:48:33
%S A156573 529,13225,444889,15108769,513249025,17435353849,592288777609,
%T A156573 20120383080625,683500735959409,23218904639535049,788759257008228025,
%U A156573 26794595833640213569,910227499086759029089,30920940373116166771225
%N A156573 a(n) = 34*a(n-1) - a(n-2) - 4232 for n > 2; a(1)=529, a(2)=13225.
%H A156573 G. C. Greubel, <a href="/A156573/b156573.txt">Table of n, a(n) for n = 1..600</a>
%H A156573 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (35,-35,1).
%F A156573 a(n) = 529*(2 + (3 - 2*sqrt(2))*(17 + 12*sqrt(2))^n + (3 + 2*sqrt(2))*(17 - 12*sqrt(2))^n)/8.
%F A156573 a(n) = 529*A008844(n).
%F A156573 G.f.: 529*x*(1 -10*x +x^2)/((1-x)*(1-34*x+x^2)). [corrected by _Klaus Brockhaus_, Sep 22 2009]
%F A156573 Limit_{n -> infinity} a(n)/a(n-1) = 17+12*sqrt(2).
%F A156573 a(n) = -529*[n=0] + (529/4) + (1587/4)*(ChebyshevU(n, 17) - 33*ChebyshevU(n-1, 17)). - _G. C. Greubel_, Jan 04 2022
%e A156573 a(3) = 34*a(2) - a(1) - 4232 = 34*13225 - 529 - 4232 = 444889.
%t A156573 LinearRecurrence[{35,-35,1}, {529,13225,444889}, 30] (* _G. C. Greubel_, Jan 04 2022 *)
%o A156573 (PARI) {m=14; v=concat([529, 13225], vector(m-2)); for(n=3, m, v[n]=34*v[n-1]-v[n-2]-4232); v}
%o A156573 (Sage)
%o A156573 def a(n): return -529*bool(n==0) + (529/4) + (3/4)*(chebyshev_U(n, 17) - 33*chebyshev_U(n-1, 17))
%o A156573 [a(n) for n in (1..30)] # _G. C. Greubel_, Jan 04 2022
%Y A156573 Second trisection of A156572.
%Y A156573 Cf. A008844, A156164 (decimal expansion of 17+12*sqrt(2)), A156574, A156575.
%K A156573 nonn,easy
%O A156573 1,1
%A A156573 _Klaus Brockhaus_, Feb 11 2009
%E A156573 Revised by _Klaus Brockhaus_, Feb 16 2009