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A210729 a(n) = a(n-1) + a(n-2) + n + 3 with n>1, a(0)=1, a(1)=2.

%I A210729 #40 Sep 08 2022 08:46:01
%S A210729 1,2,8,16,31,55,95,160,266,438,717,1169,1901,3086,5004,8108,13131,
%T A210729 21259,34411,55692,90126,145842,235993,381861,617881,999770,1617680,
%U A210729 2617480,4235191,6852703,11087927,17940664,29028626,46969326,75997989,122967353
%N A210729 a(n) = a(n-1) + a(n-2) + n + 3 with n>1, a(0)=1, a(1)=2.
%H A210729 Vincenzo Librandi, <a href="/A210729/b210729.txt">Table of n, a(n) for n = 0..200</a>
%H A210729 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-1,1).
%F A210729 G.f.: (1-x+4*x^2-3*x^3)/((1-x-x^2)*(1-x)^2).
%F A210729 a(n) = 3*Fibonacci(n+1)+2*Fibonacci(n+3)-n-6. - _Vaclav Kotesovec_, May 13 2012
%F A210729 a(n) = 2*Lucas(n+2) + Fibonacci(n+1) - (n+6). - _G. C. Greubel_, Jul 09 2019
%t A210729 Table[3*Fibonacci[n+1]+2*Fibonacci[n+3]-n-6,{n,0,40}] (* _Vaclav Kotesovec_, May 13 2012 *)
%o A210729 (Python)
%o A210729 prpr, prev = 1,2
%o A210729 for n in range(2, 99):
%o A210729     current = prev+prpr+n+3
%o A210729     print(prpr, end=',')
%o A210729     prpr = prev
%o A210729     prev = current
%o A210729 (Magma) [3*Fibonacci(n+1)+2*Fibonacci(n+3)-n-6: n in [0..40]]; // _Vincenzo Librandi_, Jul 18 2013
%o A210729 (PARI) vector(40, n, n--; f=fibonacci; 2*f(n+3)+3*f(n+1)-n-6) \\ _G. C. Greubel_, Jul 09 2019
%o A210729 (Sage) f=fibonacci; [2*f(n+3)+3*f(n+1)-n-6 for n in (0..40)] # _G. C. Greubel_, Jul 09 2019
%o A210729 (GAP) F:=Fibonacci;; List([0..40], n-> 2*F(n+3)+3*F(n+1)-n-6); # _G. C. Greubel_, Jul 09 2019
%Y A210729 Cf. A065220: a(n)=a(n-1)+a(n-2)+n-5, a(0)=1,a(1)=2 (except first 2 terms).
%Y A210729 Cf. A168043: a(n)=a(n-1)+a(n-2)+n-3, a(0)=1,a(1)=2 (except first 2 terms).
%Y A210729 Cf. A131269: a(n)=a(n-1)+a(n-2)+n-2, a(0)=1,a(1)=2.
%Y A210729 Cf. A000126: a(n)=a(n-1)+a(n-2)+n-1, a(0)=1,a(1)=2.
%Y A210729 Cf. A104161: a(n)=a(n-1)+a(n-2)+n,   a(0)=1,a(1)=2 (except the first term).
%Y A210729 Cf. A192969: a(n)=a(n-1)+a(n-2)+n+1, a(0)=1,a(1)=2.
%Y A210729 Cf. A210728: a(n)=a(n-1)+a(n-2)+n+2, a(0)=1,a(1)=2.
%Y A210729 Cf. A000032, A000045.
%K A210729 nonn,easy
%O A210729 0,2
%A A210729 _Alex Ratushnyak_, May 10 2012