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Numbers of the form (4^n + 4^(n-1) + ... + 1) + (n mod 2).
2

%I #22 Sep 08 2022 08:45:12

%S 6,21,86,341,1366,5461,21846,87381,349526,1398101,5592406,22369621,

%T 89478486,357913941,1431655766,5726623061,22906492246,91625968981,

%U 366503875926,1466015503701,5864062014806,23456248059221,93824992236886,375299968947541,1501199875790166

%N Numbers of the form (4^n + 4^(n-1) + ... + 1) + (n mod 2).

%H Colin Barker, <a href="/A088556/b088556.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,1,-4).

%F If n is even, then 4^n + ... + 1 = (4^(n+1) - 1)/3 = (2^(n+1) - 1)(2^n+1) + 1)/3. - _R. K. Guy_, Nov 17 2003

%F a(n) = 4*a(n-1) + a(n-2) - 4*a(n-3). - _Colin Barker_, Apr 02 2012

%F G.f.: x*(6-3*x-4*x^2) / ((1-x)*(1+x)*(1-4*x)). - _Colin Barker_, Apr 02 2012

%t LinearRecurrence[{4, 1, -4}, {6, 21, 86}, 50] (* _Vincenzo Librandi_, Jun 14 2015 *)

%o (PARI) trajpolypn(n1) = { for(x1=1,n1, y1 = polypn(4,x1); print1(y1",") ) }

%o polypn(n,p) = { x=n; if(p%2,y=2,y=1); for(m=1,p, y=y+x^m; ); return(y) }

%o (PARI) Vec(x*(6-3*x-4*x^2)/((1-x)*(1+x)*(1-4*x)) + O(x^30)) \\ _Colin Barker_, Jun 13 2015

%o (Magma) I:=[6,21,86]; [n le 3 select I[n] else 4*Self(n-1)+Self(n-2)-4*Self(n-3): n in [1..30]]; // _Vincenzo Librandi_, Jun 14 2015

%K nonn,easy

%O 1,1

%A _Cino Hilliard_, Nov 17 2003