A294157 Fibonacci sequence beginning 2, 8.

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).

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

Cf. A000032, A000045, A000285.

Similar sequences listed in A294116.

Sequence in context: A288824 A190042 A297293 * A082396 A195582 A071388

Adjacent sequences: A294154 A294155 A294156 * A294158 A294159 A294160

Keyword

nonn,easy

Author

Bruno Berselli, Oct 24 2017

Status

approved