A099858 A Chebyshev transform of (1+3x)/(1-3x).
1, 6, 17, 42, 109, 288, 755, 1974, 5167, 13530, 35423, 92736, 242785, 635622, 1664081, 4356618, 11405773, 29860704, 78176339, 204668310, 535828591, 1402817466, 3672623807, 9615053952, 25172538049, 65902560198, 172535142545
Offset: 0
Links
- Harvey P. Dale, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (3,-2,3,-1).
Programs
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Mathematica
LinearRecurrence[{3,-2,3,-1},{1,6,17,42},40] (* Harvey P. Dale, Apr 17 2024 *)
Formula
G.f.: (1+3*x+x^2)/((1+x^2)*(1-3*x+x^2)).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-1)^k*(6*3^(n-2*k-1)-0^(n-2*k)).
a(n) = Sum_{k=0..n} (0^k+6*Fibonacci(2*k))*cos(Pi*(n-k)/2).
a(n) = Sum_{k=0..n} A099857(k)*cos(Pi*(n-k)/2).
a(n) = 3*a(n-1)-2*a(n-2)+3*a(n-3)-a(n-4).
a(n) = (1/2)*(4*Fibonacci(2*n+2) - i^n - (-i)^n). - Ralf Stephan, Dec 04 2004
Comments