A159669 - OEIS

%I #37 Sep 26 2022 01:50:57

%S 1,29,811,22679,634201,17734949,495944371,13868707439,387827863921,

%T 10845311482349,303280893641851,8481019710489479,237165271000063561,

%U 6632146568291290229,185462938641156062851,5186330135384078469599,145031780852113041085921

%N Expansion of x*(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.

%H Vincenzo Librandi, <a href="/A159669/b159669.txt">Table of n, a(n) for n = 1..200</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (28,-1).

%F The a(j) recurrence is a(1)=1, a(2)=27, a(t+2) = 28*a(t+1) - a(t) resulting in terms 1, 27, 755, 21113, ... (A159668).

%F The b(j) recurrence is b(1)=1, b(2)=29, b(t+2) = 28*b(t+1) - b(t) resulting in terms 1, 29, 811, 22679, ... (this sequence).

%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 G.f.: x*(1+x)/(1-28*x+x^2). - _Vincenzo Librandi_, Feb 26 2014

%F a(n) = (14+sqrt(195))^(-n)*(-13-sqrt(195)+(-13+sqrt(195))*(14+sqrt(195))^(2*n))/26. - _Colin Barker_, Jul 25 2016

%F a(n) = chebyshev_U(n-1, 14) + chebyshev_U(n-2, 14) = A097311(n) + A097311(n-1). - _G. C. Greubel_, Sep 25 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=1..30); # _G. C. Greubel_, Sep 25 2022

%t CoefficientList[Series[(1+x)/(1-28x+x^2), {x, 0, 40}], x] (* _Vincenzo Librandi_, Feb 26 2014 *)

%t LinearRecurrence[{28,-1},{1,29},20] (* _Harvey P. Dale_, Jul 01 2019 *)

%o (PARI) Vec(x*(x+1)/(x^2-28*x+1) + O(x^100)) \\ _Colin Barker_, Feb 24 2014

%o (PARI) a(n) = round((14+sqrt(195))^(-n)*(-13-sqrt(195)+(-13+sqrt(195))*(14+sqrt(195))^(2*n))/26) \\ _Colin Barker_, Jul 25 2016

%o (Magma)

%o A097311:= func< n | Evaluate(ChebyshevSecond(n-1), 14) >;

%o [A097311(n+1) + A097311(n): n in [1..30]]; // _G. C. Greubel_, Sep 25 2022

%o (SageMath)

%o def A159669(n): return chebyshev_U(n-1, 14) + chebyshev_U(n-2, 14)

%o [A159669(n) for n in range(1,30)] # _G. C. Greubel_, Sep 25 2022

%Y Cf. A097311, A157456, A159668, A159673.

%K nonn,easy

%O 1,2

%A _Paul Weisenhorn_, Apr 19 2009

%E More terms and new name from _Colin Barker_, Feb 24 2014