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A084128
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a(n) = 4*a(n-1) + 4*a(n-2), a(0)=1, a(1)=2.
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14
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1, 2, 12, 56, 272, 1312, 6336, 30592, 147712, 713216, 3443712, 16627712, 80285696, 387653632, 1871757312, 9037643776, 43637604352, 210700992512, 1017354387456, 4912221519872, 23718303629312, 114522100596736, 552961616904192, 2669934870003712
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OFFSET
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0,2
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COMMENTS
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Original name was: Generalized Fibonacci sequence.
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LINKS
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FORMULA
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G.f.: (1-2*x)/(1-4*x-4*x^2).
a(n) = 4*a(n-1) + 4*a(n-2), a(0)=1, a(1)=2.
a(n) = (2 + 2*sqrt(2))^n/2 + (2 - 2*sqrt(2))^n/2.
E.g.f.: exp(2*x)*cosh(2*x*sqrt(2)).
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(4*k-2)/(x*(4*k+2) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 27 2013
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MAPLE
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a:=proc(n) option remember; if n=0 then 1 elif n=1 then 2 else
a := n -> (2*I)^n*ChebyshevT(n, -I):
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MATHEMATICA
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LinearRecurrence[{4, 4}, {1, 2}, 30] (* Harvey P. Dale, Mar 01 2018 *)
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PROG
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(PARI) a(n)=if(n<0, 0, polsym(4+4*x-x^2, n)[n+1]/2)
(Sage) [lucas_number2(n, 4, -4)/2 for n in range(0, 23)] # Zerinvary Lajos, May 14 2009
(Magma) [2^(n-1)*Evaluate(DicksonFirst(n, -1), 2): n in [0..40]]; // G. C. Greubel, Oct 13 2022
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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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