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

0,4

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).

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

Mathematica

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

Prog

(PARI) concat(0, Vec(x*(1+x)/(1-3*x^2+x^4) + O(x^50))) \\ Colin Barker, Mar 28 2016
(Magma) [Fibonacci(n)*(1+(-1)^n)/2 + Fibonacci(n+1)*(1-(-1)^n)/2: n in [0..40]]; // Vincenzo Librandi, Mar 29 2016

Crossrefs

Cf. A001906.

Sequence in context: A113166 A126872 A336102 * A095068 A248696 A021299

Adjacent sequences: A094963 A094964 A094965 * A094967 A094968 A094969

Keyword

easy,less,nonn

Author

Paul Barry, May 26 2004

Status

approved