This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A055269 #26 Jan 05 2025 19:51:36
%S A055269 1,7,30,116,437,1635,6106,22792,85065,317471,1184822,4421820,16502461,
%T A055269 61588027,229849650,857810576,3201392657,11947760055,44589647566,
%U A055269 166410830212,621053673285,2317803862931,8650161778442,32282843250840,120481211224921,449642001648847
%N A055269 a(n) = 4*a(n-1) - a(n-2) + 3 with a(0)=1, a(1)=7.
%C A055269 Also partial sums of A054491.
%D A055269 A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 122-125, 194-196.
%H A055269 G. C. Greubel, <a href="/A055269/b055269.txt">Table of n, a(n) for n = 0..1000</a>
%H A055269 I. Adler, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/7-2/adler.pdf">Three Diophantine equations - Part II</a>, Fib. Quart., 7 (1969), pp. 181-193.
%H A055269 E. I. Emerson, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/7-3/emerson.pdf">Recurrent Sequences in the Equation DQ^2=R^2+N</a>, Fib. Quart., 7 (1969), pp. 231-242.
%H A055269 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,-5,1).
%F A055269 G.f.: (1+2*x)/((1-x)*(1-4*x+x^2)).
%F A055269 a(n) = ( ( (17 - 5*(2-sqrt(3)))*(2+sqrt(3))^n + (5*(2+sqrt(3))-17)*(2-sqrt(3))^n )/(4*sqrt(3)) ) - 3/2.
%F A055269 a(n) = (5*ChebyshevU(n, 2) - 3*ChebyshevU(n-1, 2) - 3)/2. - _G. C. Greubel_, Mar 16 2020
%p A055269 A055269:= n-> simplify((5*ChebyshevU(n, 2) - 3*ChebyshevU(n-1, 2) - 3)/2); seq( A055269(n), n=0..40); # _G. C. Greubel_, Mar 16 2020
%t A055269 LinearRecurrence[{5,-5,1},{1,7,30},40] (* or *) CoefficientList[ Series[ (1+2*x)/(1-5*x+5*x^2-x^3),{x,0,40}],x] (* _Harvey P. Dale_, Dec 01 2013 *)
%t A055269 Table[(5*ChebyshevU[n, 2] -3*ChebyshevU[n-1, 2] - 3)/2, {n,0,40}] (* _G. C. Greubel_, Mar 16 2020 *)
%o A055269 (Magma) I:=[1,7,30]; [n le 3 select I[n] else 5*Self(n-1) - 5*Self(n-2) + Self(n-3): n in [1..40]]; // _G. C. Greubel_, Mar 16 2020
%o A055269 (Sage) [(5*chebyshev_U(n, 2) - 3*chebyshev_U(n-1, 2) - 3)/2 for n in (0..40)] # _G. C. Greubel_, Mar 16 2020
%Y A055269 Cf. A001834, A054491.
%K A055269 easy,nonn
%O A055269 0,2
%A A055269 _Barry E. Williams_, May 10 2000
%E A055269 Corrected by _T. D. Noe_, Nov 07 2006