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A217233
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Expansion of (1-2*x+x^2)/(1-3*x-3*x^2+x^3).
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3
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1, 1, 7, 23, 89, 329, 1231, 4591, 17137, 63953, 238679, 890759, 3324361, 12406681, 46302367, 172802783, 644908769, 2406832289, 8982420391, 33522849271, 125108976697, 466913057513, 1742543253359, 6503259955919, 24270496570321, 90578726325361
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OFFSET
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0,3
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COMMENTS
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Numbers with the property a(n)^2+a(n-1)^2 = 2*(a(n)-a(n-1)-(-1)^n)^2.
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REFERENCES
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R. C. Alperin, A family of nonlinear recurrences and their linear solutions, Fib. Q., 57:4 (2019), 318-321.
R. C. Alperin, A nonlinear recurrence and its relations to Chebyshev polynomials, Fib. Q., 58:2 (2020), 140-142.
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LINKS
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FORMULA
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G.f.: (1-x)^2/((1+x)*(1-4*x+x^2)).
a(n) = (4*(-2)^n+(1-sqrt(3))^(2*n+1)+(1+sqrt(3))^(2*n+1))/(6*2^n).
a(n) = -a(-n-1) = 3*a(n-1)+3*a(n-2)-a(n-3) = 4*a(n-1)-a(n-2)+4*(-1)^n.
a(n)+a(n-1) = A052530(n) with a(-1)=-1.
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EXAMPLE
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a(3)=23, a(2)=7: 23^2+7^2 = 2*(23-7-(-1)^3)^2 = 578;
a(6)=1231, a(5)=329: 1231^2+329^2 = 2*(1231-329-(-1)^6)^2 = 1623602.
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MATHEMATICA
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CoefficientList[Series[(1 - 2 x + x^2)/(1 - 3 x - 3 x^2 + x^3), {x, 0, 25}], x]
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PROG
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(PARI) Vec((1-2*x+x^2)/(1-3*x-3*x^2+x^3)+O(x^26))
(Magma) m:=26; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-2*x+x^2)/(1-3*x-3*x^2+x^3)));
(Maxima) makelist(coeff(taylor((1-2*x+x^2)/(1-3*x-3*x^2+x^3), x, 0, n), x, n), n, 0, 25);
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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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