OFFSET
1,1
COMMENTS
Extension of Fibonacci sequence, with the addition of natural number equal to the index.
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).
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)
EXAMPLE
a(5) = a(4)+a(3)+5 = 8+15+5 = 28.
PROG
(PARI) Vec(x*(x-2)*(x^2-x+1) / ((x^2+x-1)*(x-1)^2) + O(x^50)) \\ Colin Barker, Mar 12 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Carmine Suriano, Aug 05 2010
STATUS
approved