A210729 - OEIS

%I #40 Sep 08 2022 08:46:01

%S 1,2,8,16,31,55,95,160,266,438,717,1169,1901,3086,5004,8108,13131,

%T 21259,34411,55692,90126,145842,235993,381861,617881,999770,1617680,

%U 2617480,4235191,6852703,11087927,17940664,29028626,46969326,75997989,122967353

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

%H Vincenzo Librandi, <a href="/A210729/b210729.txt">Table of n, a(n) for n = 0..200</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-1,1).

%F G.f.: (1-x+4*x^2-3*x^3)/((1-x-x^2)*(1-x)^2).

%F a(n) = 3*Fibonacci(n+1)+2*Fibonacci(n+3)-n-6. - _Vaclav Kotesovec_, May 13 2012

%F a(n) = 2*Lucas(n+2) + Fibonacci(n+1) - (n+6). - _G. C. Greubel_, Jul 09 2019

%t Table[3*Fibonacci[n+1]+2*Fibonacci[n+3]-n-6,{n,0,40}] (* _Vaclav Kotesovec_, May 13 2012 *)

%o (Python)

%o prpr, prev = 1,2

%o for n in range(2, 99):

%o     current = prev+prpr+n+3

%o     print(prpr, end=',')

%o     prpr = prev

%o     prev = current

%o (Magma) [3*Fibonacci(n+1)+2*Fibonacci(n+3)-n-6: n in [0..40]]; // _Vincenzo Librandi_, Jul 18 2013

%o (PARI) vector(40, n, n--; f=fibonacci; 2*f(n+3)+3*f(n+1)-n-6) \\ _G. C. Greubel_, Jul 09 2019

%o (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 (GAP) F:=Fibonacci;; List([0..40], n-> 2*F(n+3)+3*F(n+1)-n-6); # _G. C. Greubel_, Jul 09 2019

%Y Cf. A065220: a(n)=a(n-1)+a(n-2)+n-5, a(0)=1,a(1)=2 (except first 2 terms).

%Y Cf. A168043: a(n)=a(n-1)+a(n-2)+n-3, a(0)=1,a(1)=2 (except first 2 terms).

%Y Cf. A131269: a(n)=a(n-1)+a(n-2)+n-2, a(0)=1,a(1)=2.

%Y Cf. A000126: a(n)=a(n-1)+a(n-2)+n-1, a(0)=1,a(1)=2.

%Y Cf. A104161: a(n)=a(n-1)+a(n-2)+n,   a(0)=1,a(1)=2 (except the first term).

%Y Cf. A192969: a(n)=a(n-1)+a(n-2)+n+1, a(0)=1,a(1)=2.

%Y Cf. A210728: a(n)=a(n-1)+a(n-2)+n+2, a(0)=1,a(1)=2.

%Y Cf. A000032, A000045.

%K nonn,easy

%O 0,2

%A _Alex Ratushnyak_, May 10 2012