A179991 Nonhomogeneous three-term sequence a(n) = a(n-1) + a(n-2) + n.

2, 3, 8, 15, 28, 49, 84, 141, 234, 385, 630, 1027, 1670, 2711, 4396, 7123, 11536, 18677, 30232, 48929, 79182, 128133, 207338, 335495, 542858, 878379, 1421264, 2299671, 3720964, 6020665, 9741660, 15762357, 25504050, 41266441, 66770526, 108037003, 174807566

Offset: 1

Author

Carmine Suriano, Aug 05 2010

Keywords

• nonn
• easy

Comments

Extension of Fibonacci sequence, with the addition of natural number equal to the index.

Examples

			a(5) = a(4)+a(3)+5 = 8+15+5 = 28.
		

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Crossrefs

Cf. A000045, A022319 (first differences).

Programs

PARI

Vec(x*(x-2)*(x^2-x+1) / ((x^2+x-1)*(x-1)^2) + O(x^50)) \\ Colin Barker, Mar 12 2017

Formula

a(n) = F(n)+sum(i; i=1 to n)+sum((F(k)*sum(j;j=1 to k-n-1); k=0 to n-4))+2F(n-3).

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

Limiting ratio a(n+1)/a(n) is Phi = 1.618034519...

a(n) = 2*A000032(n+1)-n-3. - R. J. Mathar, Aug 09 2010

From Colin Barker, Mar 12 2017: (Start)

a(n) = 2^(-n)*(-3*2^n+(1-sqrt(5))^(1+n) + (1+sqrt(5))^n + sqrt(5)*(1+sqrt(5))^n - 2^n*n).

a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4) for n>4.

(End)