A210730 - OEIS

%I #31 Sep 08 2022 08:46:01

%S 0,0,4,9,19,35,62,106,178,295,485,793,1292,2100,3408,5525,8951,14495,

%T 23466,37982,61470,99475,160969,260469,421464,681960,1103452,1785441,

%U 2888923,4674395,7563350,12237778,19801162,32038975,51840173,83879185,135719396

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

%C Deleting the 0's leaves row 4 of the convolution array A213579. - _Clark Kimberling_, Jun 20 2012

%H Bruno Berselli, <a href="/A210730/b210730.txt">Table of n, a(n) for n = 0..1000</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.: x^2*(4-3*x)/((1-x)^2*(1-x-x^2)). - _Bruno Berselli_, May 10 2012

%F a(n) = A210677(n)-1. - _Bruno Berselli_, May 10 2012

%F a(0)=0, a(1)=0, a(2)=4, a(3)=9, a(n) = 3*a(n-1)-2*a(n-2)-a(n-3)+a(n-4). - _Harvey P. Dale_, Jul 24 2013

%F a(n) = -5 + (2^(-1-n)*((1-sqrt(5))^n*(-7+5*sqrt(5)) + (1+sqrt(5))^n*(7+5*sqrt(5)))) / sqrt(5) - n. - _Colin Barker_, Mar 11 2017

%F a(n) = Fibonacci(n+3) + 3*Fibonacci(n+1) - n - 5. - _G. C. Greubel_, Jul 08 2019

%t RecurrenceTable[{a[0]==a[1]==0, a[n]==a[n-1] +a[n-2] +n+2}, a, {n, 40}] (* _Bruno Berselli_, May 10 2012 *)

%t LinearRecurrence[{3,-2,-1,1},{0,0,4,9},40] (* _Harvey P. Dale_, Jul 24 2013 *)

%t With[{F=Fibonacci}, Table[F[n+3]+2*F[n+1]-n-5, {n, 40}]] (* _G. C. Greubel_, Jul 08 2019 *)

%o (Magma) I:=[0, 0, 4, 9]; [n le 4 select I[n] else 3*Self(n-1)-2*Self(n-2)-Self(n-3)+Self(n-4): n in [1..37]]; // _Bruno Berselli_, May 10 2012

%o (Magma) F:=Fibonacci; [F(n+3)+3*F(n+1)-n-5: n in [0..40]]; # _G. C. Greubel_, Jul 08 2019

%o (PARI) concat(vector(2), Vec(x^2*(4-3*x)/((1-x)^2*(1-x-x^2)) + O(x^50))) \\ _Colin Barker_, Mar 11 2017

%o (PARI) vector(40, n, n--; f=fibonacci; f(n+3)+3*f(n+1)-n-5) \\ _G. C. Greubel_, Jul 08 2019

%o (Sage) f=fibonacci; [f(n+3)+3*f(n+1)-n-5 for n in (0..40)] # _G. C. Greubel_, Jul 08 2019

%o (GAP) F:=Fibonacci;; List([0..40], n-> F(n+3)+3*F(n+1)-n-5) # _G. C. Greubel_, Jul 08 2019

%Y Cf. A033818: a(n)=a(n-1)+a(n-2)+n-5, a(0)=a(1)=0 (except first 2 terms and sign).

%Y Cf. A002062: a(n)=a(n-1)+a(n-2)+n-4, a(0)=a(1)=0 (except the first term and sign).

%Y Cf. A065220: a(n)=a(n-1)+a(n-2)+n-3, a(0)=a(1)=0.

%Y Cf. A001924: a(n)=a(n-1)+a(n-2)+n-1, a(0)=a(1)=0 (except the first term).

%Y Cf. A023548: a(n)=a(n-1)+a(n-2)+n,   a(0)=a(1)=0 (except first 2 terms).

%Y Cf. A023552: a(n)=a(n-1)+a(n-2)+n+1, a(0)=a(1)=0 (except first 2 terms).

%Y Cf. A210731: a(n)=a(n-1)+a(n-2)+n+3, a(0)=a(1)=0.

%Y Cf. A000045, A213579.

%K nonn,easy

%O 0,3

%A _Alex Ratushnyak_, May 10 2012