A193744 Partial sum of Perrin numbers.

3, 3, 5, 8, 10, 15, 20, 27, 37, 49, 66, 88, 117, 156, 207, 275, 365, 484, 642, 851, 1128, 1495, 1981, 2625, 3478, 4608, 6105, 8088, 10715, 14195, 18805, 24912, 33002, 43719, 57916, 76723, 101637, 134641, 178362, 236280, 313005, 414644, 549287, 727651, 963933, 1276940, 1691586

Offset: 0

Author

Francesco Daddi, Aug 04 2011

Keywords

• nonn
• easy

Examples

			For n=2, a(2)=Perrin(0)+Perrin(1)+Perrin(2)=3+0+2=5.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1).

Crossrefs

Cf. A001608.

Programs

Maple

perrin[0]:=3: perrin[1]:=0: perrin[2]:=2: a[0]:=3: a[1]:=3: a[2]:=5:  for n from 0 to 100 do perrin[n]:=perrin[n-2]+perrin[n-3]: a[n]:=a[n-1]+perrin[n]: end do;

Mathematica

LinearRecurrence[{0, 1, 1}, {3, 0, 2}, {6, 52}] - 2 (* Alonso del Arte, Aug 05 2011, based on Harvey P. Dale's program for A001608 *)
LinearRecurrence[{1, 1, 0, -1},{3, 3, 5, 8},47] (* Ray Chandler, Aug 03 2015 *)

Formula

a(n) = Perrin(n+5)-2.

a(n) = r1^(n+5)+r2^(n+5)+r3^(n+5)-2, where r1, r2, r3 are the three roots of x^3-x-1 = 0.

G.f.: (3 - x^2)/(1 - x^2 - x^3)/(1-x) = (3 - x^2) / (1 - x - x^2 + x^4). a(n) = a(n-1) + a(n-2) - a(n-4) for n > 2. - Franklin T. Adams-Watters, Aug 05 2011