A030119 a(n) = a(n-1) + a(n-2) + n, a(0) = a(1) = 1.

1, 1, 4, 8, 16, 29, 51, 87, 146, 242, 398, 651, 1061, 1725, 2800, 4540, 7356, 11913, 19287, 31219, 50526, 81766, 132314, 214103, 346441, 560569, 907036, 1467632, 2374696, 3842357, 6217083, 10059471, 16276586, 26336090, 42612710, 68948835, 111561581

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..400

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

Formula

Periodic mod 6.

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

a(n) = Lucas(n+2) + Fibonacci(n+1) - (n+3).

a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4); a(0)=1, a(1)=1, a(2)=4, a(3)=8. - Harvey P. Dale, Nov 06 2011

a(n) = -3 + (2^(-n)*((1-sqrt(5))^n*(-3+2*sqrt(5)) + (1+sqrt(5))^n*(3+2*sqrt(5)))) / sqrt(5) - n. - Colin Barker, Mar 11 2017

Mathematica

RecurrenceTable[{a[0]==a[1]==1, a[n]==a[n-1]+a[n-2]+n}, a, {n, 40}] (* or *) LinearRecurrence[{3, -2, -1, 1}, {1, 1, 4, 8}, 40] (* Harvey P. Dale, Nov 06 2011 *)

Prog

(Magma) [Lucas(n+2) + Fibonacci(n+1) - (n+3) : n in [0..40]]; // Vincenzo Librandi, Nov 16 2011
(PARI) Vec((1-2*x+3*x^2-x^3)/((1-x-x^2)*(1-x)^2) + O(x^40)) \\ Colin Barker, Mar 11 2017
(PARI) vector(40, n, n--; f=fibonacci; f(n+3)+2*f(n+1)-n-3) \\ G. C. Greubel, Jul 24 2019
(Sage) f=fibonacci; [f(n+3)+2*f(n+1)-n-3 for n in (0..40)] # G. C. Greubel, Jul 24 2019
(GAP) F:=Fibonacci;; List([0..40], n-> F(n+3)+2*F(n+1)-n-3); # G. C. Greubel, Jul 24 2019

Crossrefs

Cf. A000032, A000045.

Sequence in context: A260515 A301148 A302508 * A034451 A099992 A301149

Adjacent sequences: A030116 A030117 A030118 * A030120 A030121 A030122

Keyword

nonn,easy

Author

Dragan Stevanovic

Extensions

Description corrected and sequence extended by Erich Friedman

Status

approved