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A048745 Partial sums of A048654.

%I A048745 #29 Sep 08 2022 08:44:57
%S A048745 1,5,14,36,89,217,526,1272,3073,7421,17918,43260,104441,252145,608734,
%T A048745 1469616,3547969,8565557,20679086,49923732,120526553,290976841,
%U A048745 702480238,1695937320,4094354881,9884647085,23863649054,57611945196,139087539449,335787024097
%N A048745 Partial sums of A048654.
%H A048745 G. C. Greubel, <a href="/A048745/b048745.txt">Table of n, a(n) for n = 0..1000</a>
%H A048745 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-1).
%F A048745 a(n) = 2*a(n-1) + a(n-2) + 3, a(0)=1, a(1)=5.
%F A048745 a(n) = ( ((4+(5/2)*sqrt(2))*(1+sqrt(2))^n - (4-(5/2)*sqrt(2))*(1-sqrt(2))^n)/ 2*sqrt(2) ) - 3/2.
%F A048745 G.f.: (1+2*x)/((1-x)*(1-2*x-x^2)). - _Paul D. Hanna_, Feb 22 2005
%F A048745 a(n) = 3*a(n-1) - a(n-2) - a(n-3), n>2, a(0)=1, a(1)=5, a(2)=14. - _Philippe Deléham_, Dec 16 2008
%F A048745 2*a(n) = A135532(n+2) - 3. - _R. J. Mathar_, Mar 06 2013
%F A048745 a(n) = (1/2)*( 5*P(n+1) + 3*P(n) - 3), where P(n) = A000129(n). - _G. C. Greubel_, May 23 2021
%t A048745 t={1,5}; Do[AppendTo[t, t[[-2]] + 2*t[[-1]] + 3], {n,40}]; t (* _Vladimir Joseph Stephan Orlovsky_, Jan 27 2012 *)
%t A048745 Accumulate[LinearRecurrence[{2,1},{1,4},30]] (* or *) LinearRecurrence[{3,-1,-1},{1,5,14},30] (* _Harvey P. Dale_, Aug 03 2020 *)
%o A048745 (PARI) a(n)=polcoeff((1+2*x)/(1-3*x+x^2+x^3)+x*O(x^n),n) \\ _Paul D. Hanna_
%o A048745 (Magma) I:=[1,5,14]; [n le 3 select I[n] else 3*Self(n-1) -Self(n-2) -Self(n-3): n in [1..31]]; // _G. C. Greubel_, May 23 2021
%o A048745 (Sage) [(5*lucas_number1(n+1,2,-1) + 3*lucas_number1(n,2,-1) -3)/2 for n in (0..30)] # _G. C. Greubel_, May 23 2021
%Y A048745 Cf. A000129, A005409, A098790.
%K A048745 easy,nonn
%O A048745 0,2
%A A048745 _Barry E. Williams_