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a(n) = a(n-2) - 2a(n-3) + a(n-4) for n>3, with a(0)=2, a(1)=0, a(2)=1, a(3)=-1, a sequence related to Pellian numbers.
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%I #16 Sep 19 2017 03:24:56

%S 2,0,1,-1,3,-3,6,-10,15,-25,41,-65,106,-172,277,-449,727,-1175,1902,

%T -3078,4979,-8057,13037,-21093,34130,-55224,89353,-144577,233931,

%U -378507,612438,-990946,1603383,-2594329,4197713,-6792041,10989754,-17781796,28771549,-46553345

%N a(n) = a(n-2) - 2a(n-3) + a(n-4) for n>3, with a(0)=2, a(1)=0, a(2)=1, a(3)=-1, a sequence related to Pellian numbers.

%C Successive differences begin:

%C 2, 0, 1, -1, 3, -3, 6, -10, 15, -25, ... = a(n)

%C -2, 1, -2, 4, -6, 9, -16, 25, -40, 66, ... = b(n)

%C 3, -3, 6, -10, 15, -25, 41, -65, 106, -172, ... = a(n+4)

%C -6, 9, -16, 25, -40, 66, -106, 171, -278, 449, ... = b(n+4)

%C 15, -25, 41, -65, 106, -172, 277, -449, 727, -1175, ... = a(n+8)

%C ...

%C Main diagonal [2] 1, 6, 25, 106, 449, ... (omitting first term) is A048875 (Pellian numbers with second term 6).

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,-2,1).

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

%F a(n) = A291660(-n) (negative indices computed using A291660 sequence function).

%F a(n) = (1/15)*2^(n-1)*((9*sqrt(5)+30)/(1+sqrt(5))^n + (30-9*sqrt(5))/(1- sqrt(5))^n - 5*sqrt(3)*2^(1-n)*sin(n*Pi/3)).

%t LinearRecurrence[{0, 1, -2, 1}, {2, 0, 1, -1}, 40]

%o (PARI) x='x+O('x^99); Vec((2-x^2+3*x^3)/(1-x^2+2*x^3-x^4)) \\ _Altug Alkan_, Sep 18 2017

%Y Cf. A048875, A291660.

%K sign

%O 0,1

%A _Jean-François Alcover_ and _Paul Curtz_, Sep 18 2017