A227104 a(0)=-1, a(1)=3; a(n+2) = a(n+1) + a(n) + 2*A057078(n+1).

-1, 3, 2, 3, 7, 10, 15, 27, 42, 67, 111, 178, 287, 467, 754, 1219, 1975, 3194, 5167, 8363, 13530, 21891, 35423, 57314, 92735, 150051, 242786, 392835, 635623, 1028458, 1664079, 2692539, 4356618, 7049155, 11405775, 18454930, 29860703, 48315635, 78176338, 126491971

Offset: 0

Author

Paul Curtz, Jul 01 2013

Keywords

• sign
• easy

Comments

a(n+1)/a(n) tends to A001622 (the golden ratio) as n -> infinity.

a(n) and its differences:

. -1, 3, 2, 3, 7, 10, 15, 27, 42,

. 4, -1, 1, 4, 3, 5, 12, 15, 25,

. -5, 2, 3, -1, 2, 7, 3, 10, 19,

. 7, 1, -4, 3, 5, -4, 7, 9, 4,

. -6, -5, 7, 2, -9, 11, 2, -5, 15,

. 1, 12, -5, -11, 20, -9, -7, 20, -5,

. 11, -17, -6, 31, -29, 2, 27, -25, 2,

. -28, 11, 37, -60, 31, 25, -52, 27, 29,

. 39, 26, -97, 91, -6, -77, 79, 2, -81.

Main diagonal: -(-1)^floor(n/2)*A108411(n).

Examples

			a(6) = 2*F(6)-1 = 2*8-1 = 15; a(7) = 2*F(7)+1 = 2*13+1 = 27; a(8) = 2*F(8) = 2*21 = 42.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (0,1,2,1).

Crossrefs

Cf. A000045.

Programs

Mathematica

a[n_] := (m = Mod[n, 3]; 2*Fibonacci[n] - (3*m - 1)*(m - 2)/2); Table[a[n], {n, 0, 39}]  (* Jean-François Alcover, Jul 02 2013 *)

Formula

a(3n) = 2*F(3n)-1, a(3n+1) = 2*F(3n+1)+1, a(3n+2) = 2*F(3n+2), where F=A000045.

a(n+3) = a(n) + 4*F(n+1).

a(n) = A226328(n) + 1 for n>1.

a(n) = a(n-1) + a(n-2) + a(n-3) - a(n-4) - a(n-5) and many others by telescoping the fundamental recurrence.

G.f.: -(1-3*x-3*x^2-2*x^3) / ( (1-x-x^2)*(1+x+x^2) ). [Bruno Berselli, Jul 02 2013]

a(n) = a(n-2) + 2*a(n-3) - a(n-4). [Bruno Berselli, Jul 02 2013]

Extensions

Edited by Bruno Berselli, Jul 02 2013