A131068 - OEIS

%I #22 Feb 08 2022 19:13:44

%S 1,2,10,32,82,188,406,848,1738,3524,7102,14264,28594,57260,114598,

%T 229280,458650,917396,1834894,3669896,7339906,14679932,29359990,

%U 58720112,117440362,234880868,469761886,939523928,1879048018,3758096204

%N Binomial transform of [1, 1, 7, 7, 7, ...].

%C Row sums of triangle A131067. - _Emeric Deutsch_, Jun 20 2007

%H Vincenzo Librandi, <a href="/A131068/b131068.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 From _Emeric Deutsch_, Jun 20 2007: (Start)

%F a(n) = 7*2^n - 6*(n + 1).

%F G.f.: (1 - 2*x + 7*x^2)/((1-2*x)*(1-x)^2). (End)

%F a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3). - _Vincenzo Librandi_, Jul 05 2012

%F E.g.f.: 7*exp(2*x) - 6*(1+x)*exp(x). - _G. C. Greubel_, Mar 12 2020

%e a(3) = 32 = sum of row 4 terms of triangle A131067: (1 + 15 + 15 + 1).

%e a(3) = 32 = (1, 3, 3, 1) dot (1, 1, 7, 7) = (1 + 3 + 21 + 7).

%p a := proc (n) options operator, arrow; 7*2^n-6*n-6 end proc: seq(a(n), n = 0 .. 30); # _Emeric Deutsch_, Jun 20 2007

%t Table[7*2^n -6*(n+1), {n,0,30}] (* _Vladimir Joseph Stephan Orlovsky_, Nov 10 2008; modified by _G. C. Greubel_, Mar 12 2020 *)

%t CoefficientList[Series[(1-2x+7x^2)/((1-2x)(1-x)^2),{x,0,40}],x] (* _Vincenzo Librandi_, Jul 05 2012 *)

%o (Magma) I:=[1, 2, 10]; [n le 3 select I[n] else 4*Self(n-1)-5*Self(n-2) + 2*Self(n-3): n in [1..30]]; // _Vincenzo Librandi_, Jul 05 2012

%o (PARI) vector(31, n, my(m=n-1); 7*2^m -6*(m+1) ) \\ _G. C. Greubel_, Mar 12 2020

%o (Sage) [7*2^n -6*(n+1) for n in (0..30)] # _G. C. Greubel_, Mar 12 2020

%Y Cf. A131060, A131061, A131063, A131064, A131065, A131066, A131067.

%K nonn,easy

%O 0,2

%A _Gary W. Adamson_, Jun 13 2007

%E Corrected and extended by _Emeric Deutsch_, Jun 20 2007