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A179992 Extended three term Fibonacci sequence a(n)=a(n-1)+a(n-2)+n^2. a(1)=2; a(2)=5 0
2, 5, 16, 37, 78, 151, 278, 493, 852, 1445, 2418, 4007, 6594, 10797, 17616, 28669, 46574, 75567, 122502, 198469, 321412, 520365, 842306, 1363247, 2206178, 3570101, 5777008, 9347893, 15125742, 24474535, 39601238, 64076797, 103679124 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Each term is the sum of the previous two plus the square of its index.

LINKS

Table of n, a(n) for n=1..33.

Index entries for linear recurrences with constant coefficients, signature (4,-5,1,2,-1).

FORMULA

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

G.f.: x*(x^4-4*x^3+6*x^2-3*x+2)/((1-x-x^2)*(1-x)^3)

Limiting ratio a(n+1)/a(n)=Phi=1.618038...

a(n) = 2*A022095(n+2)-6*n-13-n^2. [From R. J. Mathar, Aug 06 2010]

a(n)-4*a(n-1)+5*a(n-2)-a(n-3)-2*a(n-4)+a(n-5) = 0 with n>5. [From Bruno Berselli, Aug 25 2010]

EXAMPLE

a(5)=a(4)+a(3)+5^2=16+37+25=78

CROSSREFS

Cf. A000045, A179991

Cf. A160536, A163250. [From Bruno Berselli, Aug 25 2010]

Sequence in context: A053683 A305876 A082085 * A054971 A124720 A188947

Adjacent sequences:  A179989 A179990 A179991 * A179993 A179994 A179995

KEYWORD

nonn

AUTHOR

Carmine Suriano, Aug 05 2010

EXTENSIONS

Denominator of the g.f. replaced with product of factors by Bruno Berselli, Aug 25 2010

Multiplied g.f. with x to match the offset - R. J. Mathar, Oct 18 2010

STATUS

approved

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Last modified March 28 11:00 EDT 2020. Contains 333083 sequences. (Running on oeis4.)