A135343 - OEIS

%I #14 Oct 11 2016 06:24:52

%S 1,3,12,51,205,820,3277,13107,52428,209715,838861,3355444,13421773,

%T 53687091,214748364,858993459,3435973837,13743895348,54975581389,

%U 219902325555,879609302220,3518437208883,14073748835533,56294995342132,225179981368525,900719925474099

%N a(n) = 3*a(n-1) + 4*a(n-2) - a(n-3) + 3*a(n-4) + 4*a(n-5).

%C See A129339 comments. Third sequence after b(n) = 3*b(n-1) - 3*b(n-2) + 2*b(n-3) and c(n) = 3*c(n-1) - c(n-3) + 3*c(n-4). The first is for every sequence identical to its third differences. What characterizes the two others?

%H G. C. Greubel, <a href="/A135343/b135343.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3, 4, -1, 3, 4).

%F a(n+1) - 4*a(n) = hexaperiodic -1, 0, 3, 1, 0, -3.

%F a(n) = (1/15)*( 3*4^(n+1) - 2*(-1)^n + 5*cos(Pi*n/3) - 5*sqrt(3)*cos(Pi*n/3) ). - _Richard Choulet_, Jan 04 2008

%F G.f.: (1-x+4*x^3) / ((1+x)*(1-4*x)*(1-x+x^2)). - _Colin Barker_, Oct 11 2016

%t LinearRecurrence[{3,4,-1,3,4},{1,3,12,51,205},30] (* _Harvey P. Dale_, Jun 03 2013 *)

%o (PARI) Vec((1-x+4*x^3)/((1+x)*(1-4*x)*(1-x+x^2)) + O(x^30)) \\ _Colin Barker_, Oct 11 2016

%Y Cf. A129339, A135345.

%K nonn,easy

%O 0,2

%A _Paul Curtz_, Dec 05 2007

%E More terms from _Harvey P. Dale_, Jun 03 2013

%E Removed incorrect formula, _Joerg Arndt_, Oct 11 2016