OFFSET
0,1
REFERENCES
Steven Vajda, Fibonacci and Lucas Numbers, and the Golden Section: Theory and Applications, Dover Publications (2008), page 24 (formula 8).
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (1,1).
FORMULA
G.f.: 2*(1 + 3*x)/(1 - x - x^2).
a(n) = a(n-1) + a(n-2).
a(n) = 2*A000285(n).
Let g(r,s;n) be the n-th generalized Fibonacci number with initial values r, s. We have:
a(n) = Lucas(n) + g(0,7;n), see A022090;
a(n) = Fibonacci(n) + g(2,7;n), see A022113;
a(n) = 2*g(1,8;n) - g(0,8;n);
a(n) = g(1,k;n) + g(1,8-k;n) for all k in Z.
a(h+k) = a(h)*Fibonacci(k-1) + a(h+1)*Fibonacci(k) for all h, k in Z (see S. Vajda in References section). For h=0 and k=n:
a(n) = 2*Fibonacci(n-1) + 8*Fibonacci(n).
Sum_{j=0..n} a(j) = a(n+2) - 8.
a(n) = (2^(-n)*((1-sqrt(5))^n*(-7+sqrt(5)) + (1+sqrt(5))^n*(7+sqrt(5)))) / sqrt(5). - Colin Barker, Oct 25 2017
MAPLE
f:= gfun:-rectoproc({a(n)=a(n-1)+a(n-2), a(0)=2, a(1)=8}, a(n), remember):
map(f, [$0..100]); # Robert Israel, Oct 24 2017
MATHEMATICA
LinearRecurrence[{1, 1}, {2, 8}, 40]
PROG
(Magma) a0:=2; a1:=8; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..40]];
(PARI) Vec(2*(1 + 3*x)/(1 - x - x^2) + O(x^40)) \\ Colin Barker, Oct 25 2017
(Sage)
a = BinaryRecurrenceSequence(1, 1, 2, 8)
print([a(n) for n in range(38)]) # Peter Luschny, Oct 25 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Bruno Berselli, Oct 24 2017
STATUS
approved