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A159675 Expansion of x*(1 + x)/(1 - 32*x + x^2).

%I A159675 #28 Sep 26 2022 01:50:51
%S A159675 1,33,1055,33727,1078209,34468961,1101928543,35227244415,
%T A159675 1126169892737,36002209323169,1150944528448671,36794222701034303,
%U A159675 1176264181904649025,37603659598247734497,1202140842962022854879,38430903315186483621631,1228586765243005453037313
%N A159675 Expansion of x*(1 + x)/(1 - 32*x + x^2).
%C A159675 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 integer numbers.
%H A159675 Colin Barker, <a href="/A159675/b159675.txt">Table of n, a(n) for n = 1..650</a>
%H A159675 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (32,-1).
%F A159675 The a(j) recurrence is a(1)=1; a(2)=31; a(t+2)=32*a(t+1)-a(t) resulting in terms 1, 31, 991, 31681... (A159674).
%F A159675 The b(j) recurrence is b(1)=1; b(2)=33; b(t+2)=32*b(t+1)-b(t) resulting in terms 1, 33, 1055, 33727... (this sequence).
%F A159675 The n(j) recurrence is n(0)=n(1)=0; n(2)=64; n(t+3)=1023*(n(t+2)-n(t+1))+n(t) resulting in terms 0, 0, 64, 65472, 66912384... (A159677).
%F A159675 G.f.: x*(1 + x)/(1 - 32*x + x^2). - _Harvey P. Dale_, Apr 22 2011
%F A159675 a(n) = (16+sqrt(255))^(-n)*(-15 - sqrt(255) + (-15 + sqrt(255))*(16 + sqrt(255))^(2*n))/30. - _Colin Barker_, Jul 25 2016
%F A159675 a(n) = ChebyshevU(n-1, 16) + ChebyshevU(n-2, 16). - _G. C. Greubel_, Sep 25 2022
%p A159675 for a from 1 by 2 to 100000 do b:=sqrt((17*a*a-2)/15): if (trunc(b)=b) then
%p A159675 n:=(a*a-1)/15: La:=[op(La),a]:Lb:=[op(Lb),b]:Ln:=[op(Ln),n]: endif: enddo:
%t A159675 LinearRecurrence[{32,-1},{1,33},20] (* or *)
%t A159675 CoefficientList[Series[(1+x)/(1-32 x+x^2),{x,0,20}], x] (* _Harvey P. Dale_, Apr 22 2011 *)
%o A159675 (PARI) Vec(x*(1+x)/(1-32*x+x^2) + O(x^20)) \\ _Colin Barker_, Feb 24 2014
%o A159675 (PARI) a(n) = round((16+sqrt(255))^(-n)*(-15-sqrt(255)+(-15+sqrt(255))*(16+sqrt(255))^(2*n))/30) \\ _Colin Barker_, Jul 25 2016
%o A159675 (Magma) [n le 2 select (33)^(n-1) else 32*Self(n-1) -Self(n-2): n in [1..30]]; // _G. C. Greubel_, Sep 25 2022
%o A159675 (SageMath)
%o A159675 def A159675(n): return chebyshev_U(n-1, 16) + chebyshev_U(n-2, 16)
%o A159675 [A159675(n) for n in range(1,30)] # _G. C. Greubel_, Sep 25 2022
%Y A159675 Cf. A157456, A159674, A159677.
%K A159675 nonn,easy
%O A159675 1,2
%A A159675 _Paul Weisenhorn_, Apr 19 2009
%E A159675 More terms from _Harvey P. Dale_, Apr 22 2011
%E A159675 New name from _Colin Barker_, Feb 24 2014