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A054783
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(n^2)-th Fibonacci number.
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10
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0, 1, 3, 34, 987, 75025, 14930352, 7778742049, 10610209857723, 37889062373143906, 354224848179261915075, 8670007398507948658051921, 555565404224292694404015791808, 93202207781383214849429075266681969, 40934782466626840596168752972961528246147
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OFFSET
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0,3
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COMMENTS
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The sequence (5*a(n+1))_{n>=1} = (5, 15, 170, 4935, ...) is realizable in the sense that there is a self-map on a set T:X->X with the property that a(n) = #{x in X:T^nx=x} for all n >= 1. This is the simplest illustrative example of two different phenomena. The Fibonacci sequence sampled along an odd power cannot be made realizable after multiplication by a constant; the Fibonacci sequence sampled along an even power becomes realizable after multiplication by 5 (the discriminant of the sequence). This is now known to be an instance of a more general phenomenon in the following sense. If (a(n)) is a linear recurrence sequence whose characteristic polynomial F has simple zeros then the sequence (Ma(n^s)) satisfies the Dold congruence, where M=|discriminant(F)| and s is an integer multiple of the exponent of the Galois group of the splitting field of F over the rationals. Under an additional hypothesis on the signs of the coefficients of F, the sequence (Ma(n^s)) is realizable. - Thomas Ward, May 06 2022
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LINKS
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FORMULA
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MAPLE
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a:= n-> (<<0|1>, <1|1>>^(n^2))[1, 2]:
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MATHEMATICA
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PROG
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CROSSREFS
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Cf. A341617 shows a similar property for the Stirling numbers of the second kind.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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