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1, 4, 1, 144, 361, 484, 19321, 28224, 128881, 2427364, 1745041, 26501904, 285643801, 64995844, 4675961161, 31354493184, 149793121, 741117817924, 3178942795681, 545370434064, 107989070784841, 292105630845604
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OFFSET
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1,2
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COMMENTS
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The g.f. is an example of a rational function with nonnegative integer coefficients that is not N-rational.
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LINKS
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FORMULA
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G.f.: x*(1+5*x)/(1+x-5*x^2-125*x^3).
a(n) = (1/16)*( 2*5^n - (1+2*i)^(2*n) - (1-2*i)^(2*n) ) = (1/4)*( Im(1+2*i)^n )^2 = (1/4) * 5^n * sin(n*arctan(2))^2.
a(n) = (1/8)*5^n*(1 - ChebyshevU(n, -3/5) - (3/5)*ChebyshevU(n-1, -3/5)).
a(n) = (1/8)*( 5^n - (-1)^n*A066771(n) ).
E.g.f.: (1/8)*exp(-3*x)*(exp(8*x) - cos(4*x)). (End)
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MATHEMATICA
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LinearRecurrence[{-1, 5, 125}, {1, 4, 1}, 40] (* G. C. Greubel, Jan 11 2024 *)
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PROG
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(PARI) Vec((x+5*x^2)/(1+x-5*x^2-125*x^3) + O(x^30)) \\ Michel Marcus, Aug 28 2015
(Magma) I:=[1, 4, 1]; [n le 3 select I[n] else -Self(n-1) +5*Self(n-2) +125*Self(n-3): n in [1..41]]; // G. C. Greubel, Jan 11 2024
(SageMath)
@CachedFunction
if (n<4): return (0, 1, 4, 1)[n]
else: return -a(n-1) + 5*a(n-2) + 125*a(n-3)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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