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A210731 a(n) = a(n-1) + a(n-2) + n + 3 with n>1, a(0) = a(1) = 0.

%I A210731 #30 Dec 31 2024 11:14:14
%S A210731 0,0,5,11,23,42,74,126,211,349,573,936,1524,2476,4017,6511,10547,
%T A210731 17078,27646,44746,72415,117185,189625,306836,496488,803352,1299869,
%U A210731 2103251,3403151,5506434,8909618,14416086,23325739,37741861,61067637,98809536
%N A210731 a(n) = a(n-1) + a(n-2) + n + 3 with n>1, a(0) = a(1) = 0.
%H A210731 G. C. Greubel, <a href="/A210731/b210731.txt">Table of n, a(n) for n = 0..1000</a>
%H A210731 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-1,1).
%F A210731 From _Colin Barker_, Jun 29 2012: (Start)
%F A210731 a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).
%F A210731 G.f.: x^2*(5-4*x)/((1-x)^2*(1-x-x^2)). (End)
%F A210731 a(n) = Fibonacci(n+3) + 4*Fibonacci(n+1) - (n+6). - _G. C. Greubel_, Jul 09 2019
%t A210731 With[{F = Fibonacci}, Table[F[n+3]+4*F[n+1]-n-6, {n,0,40}]] (* _G. C. Greubel_, Jul 09 2019 *)
%t A210731 nxt[{n_,a_,b_}]:={n+1,b,a+b+n+4}; NestList[nxt,{1,0,0},40][[;;,2]] (* or *) LinearRecurrence[{3,-2,-1,1},{0,0,5,11},40] (* _Harvey P. Dale_, Dec 30 2024 *)
%o A210731 (PARI) vector(40, n, n--; f=fibonacci; f(n+3)+4*f(n+1)-n-6) \\ _G. C. Greubel_, Jul 09 2019
%o A210731 (Magma) F:=Fibonacci; [F(n+3)+4*F(n+1)-n-6: n in [0..40]]; // _G. C. Greubel_, Jul 09 2019
%o A210731 (Sage) f=fibonacci; [f(n+3)+4*f(n+1)-n-6 for n in (0..40)] # _G. C. Greubel_, Jul 09 2019
%o A210731 (GAP) F:=Fibonacci;; List([0..40], n-> F(n+3)+4*F(n+1)-n-6); # _G. C. Greubel_, Jul 09 2019
%Y A210731 Cf. A033818: a(n)=a(n-1)+a(n-2)+n-5, a(0)=a(1)=0 (except first 2 terms and sign).
%Y A210731 Cf. A002062: a(n)=a(n-1)+a(n-2)+n-4, a(0)=a(1)=0 (except the first term and sign).
%Y A210731 Cf. A065220: a(n)=a(n-1)+a(n-2)+n-3, a(0)=a(1)=0.
%Y A210731 Cf. A001924: a(n)=a(n-1)+a(n-2)+n-1, a(0)=a(1)=0 (except the first term).
%Y A210731 Cf. A023548: a(n)=a(n-1)+a(n-2)+n,   a(0)=a(1)=0 (except first 2 terms).
%Y A210731 Cf. A023552: a(n)=a(n-1)+a(n-2)+n+1, a(0)=a(1)=0 (except first 2 terms).
%Y A210731 Cf. A210730: a(n)=a(n-1)+a(n-2)+n+2, a(0)=a(1)=0.
%K A210731 nonn,easy
%O A210731 0,3
%A A210731 _Alex Ratushnyak_, May 10 2012