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A294157
Fibonacci sequence beginning 2, 8.
2
2, 8, 10, 18, 28, 46, 74, 120, 194, 314, 508, 822, 1330, 2152, 3482, 5634, 9116, 14750, 23866, 38616, 62482, 101098, 163580, 264678, 428258, 692936, 1121194, 1814130, 2935324, 4749454, 7684778, 12434232, 20119010, 32553242, 52672252, 85225494, 137897746, 223123240
OFFSET
0,1
REFERENCES
Steven Vajda, Fibonacci and Lucas Numbers, and the Golden Section: Theory and Applications, Dover Publications (2008), page 24 (formula 8).
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
Similar sequences listed in A294116.
Sequence in context: A288824 A190042 A297293 * A082396 A195582 A071388
KEYWORD
nonn,easy
AUTHOR
Bruno Berselli, Oct 24 2017
STATUS
approved