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a(n) = (3^n +1)/2 + 2^n.
4

%I #19 Sep 08 2022 08:45:15

%S 2,4,9,22,57,154,429,1222,3537,10354,30549,90622,269817,805354,

%T 2407869,7207222,21588897,64701154,193972389,581655022,1744440777,

%U 5232273754,15694724109,47079978022,141231545457,423677859154,1271000023029

%N a(n) = (3^n +1)/2 + 2^n.

%C Let b(0)=1, b(n) = A005578(n-1) = {1,1,2,3,6,11,22,43,86,171,342, ...} then a(n) = Sum_{k=0..n+1} C(n+1,k)*b(k).

%C Binomial transform of A135351. - _R. J. Mathar_, Aug 05 2009

%H G. C. Greubel, <a href="/A099754/b099754.txt">Table of n, a(n) for n = 0..1000</a>

%H Yilmaz Simsek, <a href="http://arxiv.org/abs/1604.05601">New families of special numbers for computing negative order Euler numbers</a>, arXiv:1604.05601 [math.NT], 2016.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-11,6).

%F a(n) = (3^n + 2^(n+1) + 1)/2.

%F G.f.: (2-8*x+7*x^2)/((1-x)*(1-2*x)*(1-3*x)). - _Jaume Oliver Lafont_, Mar 06 2009

%F a(n) = A007051(n) + A000079(n). - _Michel Marcus_, Aug 15 2013

%F E.g.f.: (exp(x) + 2*exp(2*x) + exp(3*x))/2. - _G. C. Greubel_, Sep 03 2019

%e a(6) = (3^6+1)/2 + 2^6 = 365+64 = 429.

%e a(6) = 1 + 7*1 + 21*1 + 35*2 + 35*3 + 21*6 + 7*11 + 1*22 = 429.

%p seq((3^n +2^(n+1) +1)/2, n=0..30); # _G. C. Greubel_, Sep 03 2019

%t Table[(3^n +2^(n+1) +1)/2, {n,0,30}] (* _G. C. Greubel_, Sep 03 2019 *)

%t LinearRecurrence[{6,-11,6},{2,4,9},30] (* _Harvey P. Dale_, May 23 2021 *)

%o (PARI) a(n) = (3^n+1)/2 + 2^n; \\ _Michel Marcus_, Aug 15 2013

%o (Magma) [(3^n +2^(n+1) +1)/2: n in [0..30]]; // _G. C. Greubel_, Sep 03 2019

%o (Sage) [(3^n +2^(n+1) +1)/2 for n in (0..30)] # _G. C. Greubel_, Sep 03 2019

%o (GAP) List([0..30], n-> (3^n +2^(n+1) +1)/2); # _G. C. Greubel_, Sep 03 2019

%Y Cf. A005578.

%K easy,nonn

%O 0,1

%A _Miklos Kristof_, Nov 11 2004

%E Corrected and extended by _T. D. Noe_, Nov 07 2006