A094966 Left-hand neighbors of Fibonacci numbers in Stern's diatomic series.

0, 1, 1, 3, 3, 8, 8, 21, 21, 55, 55, 144, 144, 377, 377, 987, 987, 2584, 2584, 6765, 6765, 17711, 17711, 46368, 46368, 121393, 121393, 317811, 317811, 832040, 832040, 2178309, 2178309, 5702887, 5702887, 14930352, 14930352, 39088169, 39088169

Offset: 0

Author

Paul Barry, May 26 2004

Keywords

• easy
• less
• nonn

Comments

Fibonacci(2n) repeated. a(n) is the left neighbor of Fibonacci(n+2) in A002487 and A049456. A000045(n+2) = a(n)+A094967(n).

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,3,0,-1).

Crossrefs

Cf. A001906.

Programs

Magma

[Fibonacci(n)*(1+(-1)^n)/2 + Fibonacci(n+1)*(1-(-1)^n)/2: n in [0..40]]; // Vincenzo Librandi, Mar 29 2016

Mathematica

CoefficientList[Series[x (1 + x)/(1 - 3 x^2 + x^4), {x, 0, 38}], x] (* Michael De Vlieger, Mar 28 2016 *)

PARI

concat(0, Vec(x*(1+x)/(1-3*x^2+x^4) + O(x^50))) \\ Colin Barker, Mar 28 2016

Formula

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

a(n) = Fibonacci(n)*(1+(-1)^n)/2 + Fibonacci(n+1)*(1-(-1)^n)/2.

a(n) = (2^(-2-n)*((1-sqrt(5))^n*(-3+sqrt(5)) - (-1-sqrt(5))^n*(-1+sqrt(5)) - (-1+sqrt(5))^n - sqrt(5)*(-1+sqrt(5))^n + 3*(1+sqrt(5))^n + sqrt(5)*(1+sqrt(5))^n))/sqrt(5). - Colin Barker, Mar 28 2016