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a(n) = a(n-1) + 2*a(n-2) + (-1)^n.
(Formerly M3254)
1

%I M3254 #29 Jun 28 2023 20:46:58

%S 1,1,4,5,14,23,52,97,202,395,800,1589,3190,6367,12748,25481,50978,

%T 101939,203896,407773,815566,1631111,3262244,6524465,13048954,

%U 26097883,52195792,104391557

%N a(n) = a(n-1) + 2*a(n-2) + (-1)^n.

%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 327.

%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2d edition 1994, p. 341.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0, 3, 2).

%F G.f.: x^4*(-x^2 - x - 1)/((1 + x)^2 (2 x - 1)). - corrected by _Harvey P. Dale_, Apr 22 2011

%F With offset 0: a(n) = 1/9*(7*2^n+(-1)^n*(3*n+2)); if b(1)=1, b(k) = 2*b(k-1)+(-1)^k*k, then for n>4, a(n)=b(n-4). - _Benoit Cloitre_, Oct 28 2002

%F a[0]=a[1]=1; a[2]=4; a[n]=3a[n-2]+2a[n-3] [From Harvey P. Dale, Apr 22 2011]

%t LinearRecurrence[{0,3,2},{1,1,4},41] (* or *) CoefficientList[Series[ (-x^2-x-1)/((1+x)^2 (2x-1)),{x,0,40}],x] (* _Harvey P. Dale_, Apr 22 2011 *)

%K nonn,easy

%O 4,3

%A _Simon Plouffe_ and _N. J. A. Sloane_.