A129982 Fibonacci numbers sandwiched between 1'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, 377, 1, 610, 1, 987, 1, 1597, 1, 2584, 1, 4181, 1, 6765, 1, 10946, 1, 17711, 1, 28657, 1, 46368, 1, 75025, 1, 121393, 1, 196418, 1, 317811, 1, 514229, 1, 832040, 1, 1346269, 1
Offset: 0
Examples
G.f. = 1 + x^2 + x^3 + x^4 + x^5 + x^6 + 2*x^7 + x^8 + 3*x^9 + x^10 + ...
Links
- Index entries for linear recurrences with constant coefficients, signature (0,2,0,0,0,-1).
Programs
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Maple
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
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Mathematica
a[ n_] := If[ OddQ[n], Fibonacci[ Quotient[ n, 2]], 1]; (* Michael Somos, Aug 15 2014 *)
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PARI
{a(n) = if( n%2, fibonacci( n\2), 1)}; /* Michael Somos, Aug 15 2014 */
Formula
G.f.: (1 - x^2 + x^3 - x^4 - x^5) / (1 - 2*x^2 + x^6). - Michael Somos, Aug 15 2014
a(2-n) = (-1)^(mod(n, 4) == 1) * a(n) for all n in Z. - Michael Somos, Aug 15 2014
a(2*n) = 1, a(2*n + 1) = A000045(n) for all n in Z. - Michael Somos, Aug 15 2014
a(n) = 2*a(n-2) - a(n-6) for all n in Z. - Michael Somos, Aug 15 2014
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
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
Extensions
More terms from Emeric Deutsch, Jul 09 2007
Comments