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%I #29 Mar 30 2023 05:13:42
%S 1,6,13,24,43,78,145,276,535,1050,2077,4128,8227,16422,32809,65580,
%T 131119,262194,524341,1048632,2097211,4194366,8388673,16777284,
%U 33554503,67108938,134217805,268435536,536870995,1073741910,2147483737,4294967388,8589934687,17179869282
%N a(n) = 2^(n+1) - 1 + 3*n.
%C Binomial transform of (1, 5, 2, 2, 2, ...).
%H Vincenzo Librandi, <a href="/A131833/b131833.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,2).
%F G.f.: (-1 - 2*x + 6*x^2)/((2*x - 1)*(x - 1)^2). - _R. J. Mathar_, Apr 04 2012
%F a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3). - _Vincenzo Librandi_, Jul 05 2012
%F E.g.f.: exp(x)*(2*exp(x) - 1 + 3*x). - _Stefano Spezia_, Mar 29 2023
%e a(3) = 24 = sum of row 3 terms of triangle A131832: (7 + 5 + 5 + 7).
%e a(3) = 24 = (1, 3, 3, 1) dot (1, 5, 2, 2) = (1 + 15 + 6 + 2).
%t CoefficientList[Series[(-1-2*x+6*x^2)/((2*x-1)*(x-1)^2),{x,0,40}],x] (* _Vincenzo Librandi_, Jul 05 2012 *)
%t Table[2^(n+1)-1+3n,{n,0,30}] (* or *) LinearRecurrence[{4,-5,2},{1,6,13},40] (* _Harvey P. Dale_, Nov 06 2012 *)
%o (Magma) I:=[1, 6, 13]; [n le 3 select I[n] else 4*Self(n-1)-5*Self(n-2)+2*Self(n-3): n in [1..40]]; // _Vincenzo Librandi_, Jul 05 2012
%Y Row sums of triangle A131832.
%K nonn,easy
%O 0,2
%A _Gary W. Adamson_, Jul 20 2007
%E New definition by _R. J. Mathar_, Apr 04 2012