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%I #52 Feb 16 2024 06:33:30
%S 1,27,755,21113,590409,16510339,461699083,12911063985,361048092497,
%T 10096435525931,282339146633571,7895399670214057,220788851619360025,
%U 6174192445671866643,172656599627192905979,4828210597115729500769,135017240119613233115553
%N Expansion of (1 - x)/(1 - 28*x + x^2).
%C Previous name was: The general form of the recurrences are the a(j), b(j) and n(j) solutions of the 2 equations problem: 13*n(j) + 1 = a(j)*a(j) and 15*n(j) + 1 = b(j)*b(j) with positive integer numbers.
%C Positive values of x (or y) satisfying x^2 - 28*x*y + y^2 + 26 = 0. - _Colin Barker_, Feb 23 2014
%H Vincenzo Librandi, <a href="/A159668/b159668.txt">Table of n, a(n) for n = 0..199</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (28,-1).
%F G.f.: (1 - x)/(1 - 28*x + x^2).
%F The a(j) recurrence is a(0)=1, a(1)=27, a(t+2) = 28*a(t+1) - a(t) resulting in terms 1, 27, 755, 21113, ... (this sequence).
%F The b(j) recurrence is b(0)=1, b(1)=29, b(t+2) = 28*b(t+1) - b(t) resulting in terms 1, 29, 811, 22679, ... (A159669).
%F The n(j) recurrence is n(0) = n(1) = 0, n(2) = 56, n(t+3) = 783*(n(t+2) -n(t+1)) + n(t) resulting in terms 0, 0, 56, 43848, 34289136, ... (A159673).
%F a(n) = (1/30)*(15-sqrt(195))*(1+(14+sqrt(195))^(2*n+1))/(14+sqrt(195))^n. - _Bruno Berselli_, Feb 25 2014
%F a(n) = 28*a(n-1) - a(n-2), a(0)=1, a(1)=27. - _Harvey P. Dale_, Apr 09 2014
%F a(n) = A097311(n) - A097311(n-1). - _G. C. Greubel_, Sep 26 2022
%p for a from 1 by 2 to 100000 do b:=sqrt((15*a*a-2)/13): if (trunc(b)=b) then
%p n:=(a*a-1)/13: La:=[op(La),a]:Lb:=[op(Lb),b]:Ln:=[op(Ln),n]: endif: enddo:
%p # Second program
%p seq(simplify(ChebyshevU(n,14) -ChebyshevU(n-1,14)), n=0..40); # _G. C. Greubel_, Sep 26 2022
%t CoefficientList[Series[(1-x)/(1-28x+x^2), {x,0,40}], x] (* _Vincenzo Librandi_, Feb 25 2014 *)
%t LinearRecurrence[{28,-1},{1,27},40] (* _Harvey P. Dale_, Apr 09 2014 *)
%o (PARI) Vec((-x+1)/(x^2-28*x+1) + O(x^100)) \\ _Colin Barker_, Feb 23 2014
%o (Magma) [n le 2 select 27^(n-1) else 28*Self(n-1)-Self(n-2): n in [1..20]]; // _Vincenzo Librandi_, Feb 25 2014
%o (SageMath)
%o def A159668(n): return chebyshev_U(n,14) - chebyshev_U(n-1,14)
%o [A159668(n) for n in range(40)] # _G. C. Greubel_, Sep 26 2022
%Y Cf. A097311, A157456, A159669, A159673.
%Y Cf. similar sequences listed in A238379.
%K nonn,easy
%O 0,2
%A _Paul Weisenhorn_, Apr 19 2009
%E More terms from _Colin Barker_, Feb 23 2014
%E New name and offset changed to 0 from _Joerg Arndt_, Feb 23 2014
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