cp's OEIS Frontend

A159674 Expansion of (1 - x)/(1 - 32*x + x^2).

%I A159674 #34 Sep 26 2022 01:51:05
%S A159674 1,31,991,31681,1012801,32377951,1035081631,33090234241,1057852414081,
%T A159674 33818187016351,1081124132109151,34562154040476481,
%U A159674 1104907805163138241,35322487611179947231,1129214695752595173151,36099547776471865593601,1154056314151347103822081
%N A159674 Expansion of (1 - x)/(1 - 32*x + x^2).
%C A159674 Previous name was: The general form of the recurrences are the a(j), b(j) and n(j) solutions of the 2 equations problem: 15*n(j) + 1 = a(j)*a(j) and 17*n(j) + 1 = b(j)*b(j) with positive integers.
%C A159674 Positive values of x (or y) satisfying x^2 - 32*x*y + y^2 + 30 = 0. - _Colin Barker_, Feb 24 2014
%H A159674 Vincenzo Librandi, <a href="/A159674/b159674.txt">Table of n, a(n) for n = 0..200</a>
%H A159674 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (32,-1).
%F A159674 The a(j) recurrence is: a(0)=1, a(1)=31, a(t+2) = 32*a(t+1) - a(t) resulting in terms 1, 31, 991, 31681, ... (this sequence).
%F A159674 The b(j) recurrence is: b(0)=1, b(1)=33, b(t+2) = 32*b(t+1) - b(t) resulting in terms 1, 33, 1055, 33727, ... (A159675).
%F A159674 The n(j) recurrence is: n(-1) = n(0) = 0, n(1) = 64, n(t+3) = 1023*(n(t+2) -n(t+1)) + n(t) resulting in terms 0, 0, 64, 65472, 66912384, ... (A159677).
%F A159674 a(n) = (1/34)*(17-sqrt(255))*(1+(16+sqrt(255))^(2*n+1))/(16+sqrt(255))^n. - _Bruno Berselli_, Feb 25 2014
%F A159674 a(n) = ChebyshevU(n, 16) - ChebyshevU(n-1, 16) = A029548(n) - A029548(n-1). - _G. C. Greubel_, Sep 25 2022
%p A159674 for a from 1 by 2 to 100000 do b:=sqrt((17*a*a-2)/15): if (trunc(b)=b) then
%p A159674 n:=(a*a-1)/15: La:=[op(La),a]:Lb:=[op(Lb),b]:Ln:=[op(Ln),n]: endif: enddo:
%p A159674 # Second program
%p A159674 seq(simplify(ChebyshevU(n, 16) -ChebyshevU(n-1, 16)), n=0..30); # _G. C. Greubel_, Sep 25 2022
%t A159674 CoefficientList[Series[(1-x)/(1-32*x+x^2), {x, 0, 40}], x] (* _Vincenzo Librandi_, Feb 26 2014 *)
%t A159674 LinearRecurrence[{32,-1},{1,31},30] (* _Harvey P. Dale_, Mar 21 2017 *)
%o A159674 (PARI) concat([0], Vec((-x+1)/(x^2-32*x+1) + O(x^100))) \\ _Colin Barker_, Feb 24 2014
%o A159674 (Magma)
%o A159674 A029548:= func< n | Evaluate(ChebyshevSecond(n),16) >;
%o A159674 [A029548(n+1) -A029548(n): n in [0..30]]; // _G. C. Greubel_, Sep 25 2022
%o A159674 (SageMath)
%o A159674 def A159674(n): return chebyshev_U(n, 16) - chebyshev_U(n-1, 16)
%o A159674 [A159674(n) for n in range(31)] # _G. C. Greubel_, Sep 25 2022
%Y A159674 Cf. A029548, A157456, A159675, A159677.
%Y A159674 Cf. similar sequences listed in A238379.
%K A159674 nonn,easy
%O A159674 0,2
%A A159674 _Paul Weisenhorn_, Apr 19 2009
%E A159674 More terms and new name from _Colin Barker_, Feb 24 2014
%E A159674 Set offset to 0 by _Joerg Arndt_, Feb 25 2014