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A215007
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a(n) = 7*a(n-1) - 14*a(n-2) + 7*a(n-3), a(0)=1, a(1)=3, a(2)=9.
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23
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1, 3, 9, 28, 91, 308, 1078, 3871, 14161, 52479, 196196, 737793, 2785160, 10540390, 39955041, 151615947, 575723785, 2187128524, 8311078307, 31587815308, 120069510526, 456434707519, 1735184512425, 6596692255391, 25079305566420
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OFFSET
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0,2
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COMMENTS
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The sequence {a(n)} we shall call the Berndt-type sequence of type 1 for the argument 2*Pi/7; our motivation comes from Berndt's et al. and my papers (see the first formula below, which is in agreement with the respective identities discussed in these papers).
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REFERENCES
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R. Witula, E. Hetmaniok, and D. Slota, Sums of the powers of any order roots taken from the roots of a given polynomial, Proceedings of the Fifteenth International Conference on Fibonacci Numbers and Their Applications, Eger, Hungary, 2012.
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LINKS
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FORMULA
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a(n) = (1/sqrt(7))*(cot(8*Pi/7)*(s(1))^2n + cot(4*Pi/7)*(s(4))^2n + cot(2*Pi/7)*(s(2))^2n), where s(j) := 2*sin(2Pi*j/7).
G.f.: (1-4*x+2*x^2)/(1-7*x+14*x^2-7*x^3).
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MAPLE
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seq(coeff(series((1-4*x+2*x^2)/(1-7*x+14*x^2-7*x^3), x, n+1), x, n), n = 0..30); # G. C. Greubel, Oct 03 2019
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MATHEMATICA
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LinearRecurrence[{7, -14, 7}, {1, 3, 9}, 30] (* G. C. Greubel, Feb 01 2018 *)
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PROG
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(Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!((1-4*x+2*x^2)/(1-7*x+14*x^2-7*x^3))) // G. C. Greubel, Feb 01 2018
(Sage)
P.<x> = PowerSeriesRing(ZZ, prec)
return P((1-4*x+2*x^2)/(1-7*x+14*x^2-7*x^3)).list()
(GAP) a:=[1, 3, 9];; for n in [4..30] do a[n]:=7*(a[n-1]-2*a[n-2]+a[n-3]); od; a; # G. C. Greubel, Oct 03 2019
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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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