A129982 - OEIS

%I #15 Feb 26 2019 01:37:47

%S 1,0,1,1,1,1,1,2,1,3,1,5,1,8,1,13,1,21,1,34,1,55,1,89,1,144,1,233,1,

%T 377,1,610,1,987,1,1597,1,2584,1,4181,1,6765,1,10946,1,17711,1,28657,

%U 1,46368,1,75025,1,121393,1,196418,1,317811,1,514229,1,832040,1,1346269,1

%N Fibonacci numbers sandwiched between 1's.

%C This sequence is similar to Somos-5 (A006721). - _Michael Somos_, Aug 15 2014

%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,0,0,-1).

%F G.f.: (1 - x^2 + x^3 - x^4 - x^5) / (1 - 2*x^2 + x^6). - _Michael Somos_, Aug 15 2014

%F a(2-n) = (-1)^(mod(n, 4) == 1) * a(n) for all n in Z. - _Michael Somos_, Aug 15 2014

%F a(2*n) = 1, a(2*n + 1) = A000045(n) for all n in Z. - _Michael Somos_, Aug 15 2014

%F a(n) = 2*a(n-2) - a(n-6) for all n in Z. - _Michael Somos_, Aug 15 2014

%F 0 = a(n)*a(n+5) - a(n+1)*a(n+4) - a(n+2)*a(n+3) for all even n in Z. - _Michael Somos_, Aug 15 2014

%F 0 = a(n)*a(n+5) - a(n+1)*a(n+4) + a(n+2)*a(n+3) for all odd n in Z. - _Michael Somos_, Aug 15 2014

%e G.f. = 1 + x^2 + x^3 + x^4 + x^5 + x^6 + 2*x^7 + x^8 + 3*x^9 + x^10 + ...

%p G := 1/(1-x^2)+x^3/(1-x^2-x^4); Gser := series(G, x = 0, 70); seq(coeff(Gser, x, n), n = 0 .. 65); # _Emeric Deutsch_, Jul 09 2007

%t a[ n_] := If[ OddQ[n], Fibonacci[ Quotient[ n, 2]], 1]; (* _Michael Somos_, Aug 15 2014 *)

%o (PARI) {a(n) = if( n%2, fibonacci( n\2), 1)}; /* _Michael Somos_, Aug 15 2014 */

%Y Cf. A000045, A006721.

%K nonn,easy

%O 0,8

%A _Paul Curtz_, Jun 14 2007

%E More terms from _Emeric Deutsch_, Jul 09 2007