A247563 - OEIS

%I #28 Sep 08 2022 08:46:09

%S 2,3,1,-9,-31,-57,-47,87,449,999,1201,-393,-5983,-16377,-25199,-10089,

%T 70529,251943,473713,413367,-654751,-3617721,-8234159,-10231593,

%U 2241857,47651943,133988401,211357431,98118689,-551073657,-2045695727,-3932792553,-3615594751

%N a(n) = 3*a(n-1) - 4*a(n-2) with a(0) = 2, a(1) = 3.

%C This is the Lucas sequence V_n(P, Q) = V_n(3, 4). U_n(P, Q) = U_n(3, 4) = A049072(n). ( a(n)/2 )^2 + 7*( A049072(n-1)/2 )^2 = 4^n. - _Raphie Frank_, Dec 04 2015

%H Reinhard Zumkeller, <a href="/A247563/b247563.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,-4).

%F G.f.: (2 - 3*x) / (1 - 3*x + 4*x^2).

%F a(n) = a(-n) * 4^n for all n in Z.

%F a(n) = (-1)^n * A128415(n) if n > 0.

%F a(n) = ((3 + sqrt(-7))/2)^n + ((3 - sqrt(-7))/2)^n. - _Raphie Frank_, Dec 04 2015

%e G.f. = 2 + 3*x + x^2 - 9*x^3 - 31*x^4 - 57*x^5 - 47*x^6 + 87*x^7 + ...

%t {a[0], a[1]} = {2, 3}; a[n_] := a[n] = 3 a[n - 1] - 4 a[n - 2]; Table[a@ n, {n, 0, 32}] (* _Michael De Vlieger_, Dec 04 2015 *)

%t CoefficientList[Series[(2-3*x)/(1-3*x+4*x^2), {x, 0, 60}], x] (* _G. C. Greubel_, Aug 04 2018 *)

%o (PARI) {a(n) = if( n<0, n=-n; 4^-n, 1) * polcoeff( (2 - 3*x) / (1 - 3*x + 4*x^2) + x * O(x^n), n)};

%o (Haskell)

%o a247563 n = a247563_list !! n

%o a247563_list = 2 : 3 : zipWith (-) (map (* 3) $ tail a247563_list)

%o                                    (map (* 4) a247563_list)

%o -- _Reinhard Zumkeller_, Sep 20 2014

%o (Magma) [n le 2 select n+1 else 3*Self(n-1)-4*Self(n-2): n in [1..40]]; // _Vincenzo Librandi_, Dec 05 2015

%o (Magma) m:=25; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((2 - 3*x)/(1-3*x+4*x^2)));  // _G. C. Greubel_, Aug 04 2018

%Y Cf. A128415, A049072.

%K sign,easy

%O 0,1

%A _Michael Somos_, Sep 20 2014