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

Author

Bruno Berselli, Oct 24 2017

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A000032, A000045, A000285.

Similar sequences listed in A294116.

Programs

Magma

a0:=2; a1:=8; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..40]];

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]

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

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