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A090965
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a(n) = 8*a(n-1) - 4*a(n-2), where a(0) = 1, a(1) = 4.
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12
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1, 4, 28, 208, 1552, 11584, 86464, 645376, 4817152, 35955712, 268377088, 2003193856, 14952042496, 111603564544, 833020346368, 6217748512768, 46409906716672, 346408259682304, 2585626450591744, 19299378566004736
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) = Sum_{k>=0} binomial(2*n, 2*k)*3^k = Sum_{k>=0} A086645(n, k)*3^k.
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(3*k-4)/(x*(3*k-1) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 28 2013
a(n) = Sum_{k = 0..floor(n/2)} 4^(n-2*k)*12^k*binomial(n,2*k).
a(n) = [x^n] (4*x + sqrt(1 + 12*x^2))^n.
G.f.: A(x) = 1 + x*B'(x)/B(x), where B(x) = 1/sqrt(1 - 8*x + 4*x^2) is the g.f. of A069835.
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and positive integers n and k. (End)
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MATHEMATICA
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LinearRecurrence[{8, -4}, {1, 4}, 20] (* G. C. Greubel, Feb 03 2019 *)
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PROG
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(Sage) [lucas_number2(n, 8, 4)/2 for n in range(0, 21)] # Zerinvary Lajos, Jul 08 2008
(PARI) my(x='x+O('x^20)); Vec((1-4*x)/(1-8*x+4*x^2)) \\ G. C. Greubel, Feb 03 2019
(Magma) m:=20; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-4*x)/(1-8*x+4*x^2) )); // G. C. Greubel, Feb 03 2019
(GAP) a:=[1, 4];; for n in [3..20] do a[n]:=8*a[n-1]-4*a[n-2]; od; a; # G. C. Greubel, Feb 03 2019
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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