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

0,3

Comments

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

Links

Bruno Berselli, Table of n, a(n) for n = 0..1000

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

Formula

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

Mathematica

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 *)

Prog

(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
(Magma) F:=Fibonacci; [F(n+3)+3*F(n+1)-n-5: n in [0..40]]; # G. C. Greubel, Jul 08 2019
(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
(PARI) vector(40, n, n--; f=fibonacci; f(n+3)+3*f(n+1)-n-5) \\ G. C. Greubel, Jul 08 2019
(Sage) f=fibonacci; [f(n+3)+3*f(n+1)-n-5 for n in (0..40)] # G. C. Greubel, Jul 08 2019
(GAP) F:=Fibonacci;; List([0..40], n-> F(n+3)+3*F(n+1)-n-5) # G. C. Greubel, Jul 08 2019

Crossrefs

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.

Sequence in context: A301245 A301103 A115003 * A008113 A008111 A023611

Adjacent sequences: A210727 A210728 A210729 * A210731 A210732 A210733

Keyword

nonn,easy

Author

Alex Ratushnyak, May 10 2012

Status

approved