A210677 -id:A210677 - cp's OEIS frontend

A210730 a(n) = a(n-1) + a(n-2) + n + 2 with n>1, a(0)=a(1)=0.

0, 0, 4, 9, 19, 35, 62, 106, 178, 295, 485, 793, 1292, 2100, 3408, 5525, 8951, 14495, 23466, 37982, 61470, 99475, 160969, 260469, 421464, 681960, 1103452, 1785441, 2888923, 4674395, 7563350, 12237778, 19801162, 32038975, 51840173, 83879185, 135719396

Offset: 0

Alex Ratushnyak, May 10 2012

Deleting the 0's leaves row 4 of the convolution array A213579. - Clark Kimberling, Jun 20 2012

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Cf. A033818: a(n)=a(n-1)+a(n-2)+n-5, a(0)=a(1)=0 (except first 2 terms and sign).
Cf. A002062: a(n)=a(n-1)+a(n-2)+n-4, a(0)=a(1)=0 (except the first term and sign).
Cf. A065220: a(n)=a(n-1)+a(n-2)+n-3, a(0)=a(1)=0.
Cf. A001924: a(n)=a(n-1)+a(n-2)+n-1, a(0)=a(1)=0 (except the first term).
Cf. A023548: a(n)=a(n-1)+a(n-2)+n,   a(0)=a(1)=0 (except first 2 terms).
Cf. A023552: a(n)=a(n-1)+a(n-2)+n+1, a(0)=a(1)=0 (except first 2 terms).
Cf. A210731: a(n)=a(n-1)+a(n-2)+n+3, a(0)=a(1)=0.
Cf. A000045, A213579.

F:=Fibonacci;; List([0..40], n-> F(n+3)+3*F(n+1)-n-5); # G. C. Greubel, Jul 08 2019

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

F:=Fibonacci; [F(n+3)+3*F(n+1)-n-5: n in [0..40]]; // G. C. Greubel, Jul 08 2019

RecurrenceTable[{a[0]==a[1]==0, a[n]==a[n-1] +a[n-2] +n+2}, a, {n, 40}] (* Bruno Berselli, May 10 2012 *)
LinearRecurrence[{3,-2,-1,1},{0,0,4,9},40] (* Harvey P. Dale, Jul 24 2013 *)
With[{F=Fibonacci}, Table[F[n+3]+2*F[n+1]-n-5, {n, 40}]] (* G. C. Greubel, Jul 08 2019 *)

concat(vector(2), Vec(x^2*(4-3*x)/((1-x)^2*(1-x-x^2)) + O(x^50))) \\ Colin Barker, Mar 11 2017

vector(40, n, n--; f=fibonacci; f(n+3)+3*f(n+1)-n-5) \\ G. C. Greubel, Jul 08 2019

f=fibonacci; [f(n+3)+3*f(n+1)-n-5 for n in (0..40)] # G. C. Greubel, Jul 08 2019

G.f.: x^2*(4-3*x)/((1-x)^2*(1-x-x^2)). - Bruno Berselli, May 10 2012
a(n) = A210677(n)-1. - Bruno Berselli, May 10 2012
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
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
a(n) = Fibonacci(n+3) + 3*Fibonacci(n+1) - n - 5. - G. C. Greubel, Jul 08 2019

A210678 a(n) = a(n-1)+a(n-2)+n+2, a(0)=a(1)=1.

Original entry on oeis.org

1, 1, 6, 12, 24, 43, 75, 127, 212, 350, 574, 937, 1525, 2477, 4018, 6512, 10548, 17079, 27647, 44747, 72416, 117186, 189626, 306837, 496489, 803353, 1299870, 2103252, 3403152, 5506435, 8909619, 14416087, 23325740, 37741862, 61067638, 98809537, 159877213, 258686789, 418564042

Offset: 0

Alex Ratushnyak, May 09 2012

Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Cf. A081659: a(n)=a(n-1)+a(n-2)+n-5, a(0)=a(1)=1 (except first 2 terms and sign).
Cf. A001924: a(n)=a(n-1)+a(n-2)+n-4, a(0)=a(1)=1 (except first 4 terms).
Cf. A000126: a(n)=a(n-1)+a(n-2)+n-2, a(0)=a(1)=1 (except first term).
Cf. A066982: a(n)=a(n-1)+a(n-2)+n-1, a(0)=a(1)=1.
Cf. A030119: a(n)=a(n-1)+a(n-2)+n,   a(0)=a(1)=1.
Cf. A210677: a(n)=a(n-1)+a(n-2)+n+1, a(0)=a(1)=1.

LinearRecurrence[{3,-2,-1,1},{1,1,6,12},40] (* Harvey P. Dale, Dec 10 2014 *)
nxt[{n_,a_,b_}]:={n+1,b,a+b+n+3}; NestList[nxt,{1,1,1},40][[;;,2]] (* Harvey P. Dale, Mar 19 2023 *)

From Colin Barker, Jun 30 2012: (Start)
a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).
G.f.: (1 -2*x + 5*x^2 - 3*x^3)/((1 - x)^2*(1 - x - x^2)). (End)

cp's OEIS Frontend

A210730 a(n) = a(n-1) + a(n-2) + n + 2 with n>1, a(0)=a(1)=0.

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